# Estimates for the approximation characteristics of the Nikol’skii-Besov
classes of functions with mixed smoothness in the space $\boldsymbol{B_{q,1}}$

K.V. Pozharska1,2, A.S. Romanyuk1 1Institute of Mathematics of the NAS of
Ukraine, Kyiv, Ukraine 2Faculty of Mathematics, Chemnitz University of
Technology, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Exact-order estimates are obtained for some approximation characteristics of
the classes of periodic multivariate functions with mixed smoothness (the
Nikol’skii-Besov classes $B^{\boldsymbol{r}}_{p,\theta}$) in the space
$B_{q,1}$, $1\leq p,q\leq\infty$, which norm is stronger than the
$L_{q}$-norm. It is shown, that in the multivariate case (in contrast to the
univariate) in most of the considered situations the obtained estimates differ
in order from the corresponding estimates in the space $L_{q}$.

Besides, a significant progress is made in estimates for the considered
approximation characteristics of the classes $B^{\boldsymbol{r}}_{p,\theta}$
in the space $B_{q,1}$ comparing to the known estimates in the space $L_{q}$.

###### Key words and phrases:

best approximations, Kolmogorov widths, linear widths, entropy numbers

##  1\. Introduction

In the paper, we investigate some approximation characteristics of the
Nikol’skii-Besov classes $B^{\boldsymbol{r}}_{p,\theta}$ of periodic
multivariate functions with mixed smoothness in the space $B_{q,1}$, $1\leq
p,q\leq\infty$.

The spaces $B_{q,1}$, as linear subspaces in $L_{q}$, $1\leq q\leq\infty$,
respectively, have a peculiarity in that their norm is stronger than the
$L_{q}$-norm.

Our findings complement and generalize the results, obtained in the number of
papers [3, 9, 10, 13, 15, 33, 34, 37, 44], where the authors investigate the
approximation of the Nikol’skii-Besov classes $B^{\boldsymbol{r}}_{p,\theta}$,
Sobolev classes $W^{\boldsymbol{r}}_{p,\alpha}$ of periodic multivariate
functions with mixed smoothness and some their analogs in the space $B_{q,1}$,
$q\in\\{1,\infty\\}$. Important is, that we get estimates for the
approximation characteristics of the classes $B^{\boldsymbol{r}}_{p,\theta}$
in the space $B_{q,1}$ for the case $1<q<\infty$.

A motivation for investigating the approximation problems on the respective
classes in this particular space $B_{q,1}$ is given by the fact, that the
respective problems in the space $L_{q}$ for some important cases still remain
open (see Open Problems in [8, 48]).

The paper consists of three parts.

The first part (Section 2) plays an auxiliary role. Here we introduce
necessary notation and define the considered functional classes and spaces
$B_{q,1}$, in which we measure the approximation error.

In the second part (Section 3) we obtain exact-order estimates for the
approximation of functions from the classes $B^{\boldsymbol{r}}_{p,\theta}$ in
the space $B_{q,1}$ by their step hyperbolic cross Fourier sums. Besides, we
get orders for the best approximation of the indicated function classes in the
respective space by trigonometric polynomials with harmonics from the step
hyperbolic crosses.

The third part (Section 4) is devoted to obtaining the exact-order estimates
for the Kolmogorov widths and entropy numbers of the classes
$B^{\boldsymbol{r}}_{p,\theta}$ in the space $B_{q,1}$ for some relations
between the parameters $p$ and $q$. Combining these estimates for the
Kolmogorov widths and the results of Section 3, we write down the exact-order
estimates for the linear widths of these classes in the space $B_{q,1}$.

From the results of our research, we can conclude the following.

A significant progress is made in estimates for the best approximations,
widths and entropy numbers of the classes $B^{\boldsymbol{r}}_{p,\theta}$ in
the space $B_{q,1}$ comparing to the known estimates for the corresponding
quantities in the space $L_{q}$.

We show, that the estimates for the Kolmogorov and linear widths of the
mentioned function classes in the space $B_{q,1}$ are realized by the
subspaces of trigonometric polynomials with harmonics from the step hyperbolic
crosses of the respective dimension.

We prove, that in most of the cases (for $d\geq 2$) the considered
approximation characteristics of the classes $B^{\boldsymbol{r}}_{p,\theta}$
in the space $B_{q,1}$ differ in order from the corresponding characteristics
in the space $L_{q}$.

These questions will be discussed in more details in what follows, when giving
remarks to the obtained results.

##  2\. Functional classes $B^{\boldsymbol{r}}_{p,\theta}$ and spaces
$B_{q,1}$

Let $\mathbb{R}^{d}$ be a $d$-dimensional space with elements
$\boldsymbol{x}=(x_{1},\dots,x_{d})$ and
$(\boldsymbol{x},\boldsymbol{y})=x_{1}\,y_{1}+\dots+x_{d}\,y_{d}$. By
$L_{p}(\mathbb{T}^{d})$, $\mathbb{T}^{d}=\prod^{d}_{j=1}\,[0,\,2\pi)$, we
denote the space of $2\pi$-periodic in each variable functions
$f(\boldsymbol{x})$, for which

$\displaystyle\|f\|_{p}$
$\displaystyle:=\|f\|_{{L_{p}}(\mathbb{T}^{d})}=\Big{(}(2\pi)^{-d}\int_{\mathbb{T}^{d}}|f(\boldsymbol{x})|^{p}\,d{\boldsymbol{x}}\Big{)}^{1/p}<\infty,\quad
1\leq p<\infty,$ $\displaystyle\|f\|_{\infty}$
$\displaystyle:=\|f\|_{{L_{\infty}}(\mathbb{T}^{d})}=\operatorname{ess\,sup}\limits_{{\boldsymbol{x}}\in{\mathbb{T}^{d}}}|f(\boldsymbol{x})|<\infty.$

Let further

$L^{0}_{p}:=L^{0}_{p}(\mathbb{T}^{d}):=\\{f\in
L_{p}(\mathbb{T}^{d})\colon\int^{2\pi}_{0}f(\boldsymbol{x})dx_{j}=0\ \
\text{a.e.},\,j=1,\dots,d\\}.$

For a function $f\in L^{0}_{p}$, $1\leq p\leq\infty$, we consider its first
difference in the $j$th variable with step $h\in\mathbb{R}$:

$\Delta_{h,j}f(\boldsymbol{x})=f(x_{1},\ldots,x_{j-1},x_{j}+h,x_{j+1},\ldots,x_{d})-f(\boldsymbol{x})$

and define its $l$th difference, $l\in\mathbb{N}$, by

$\Delta^{l}_{h,j}f(\boldsymbol{x})=\overbrace{\Delta_{h,j}\cdots\Delta_{h,j}}\limits^{l}f(\boldsymbol{x}).$

Assume that the vectors $\boldsymbol{k}\in\mathbb{N}^{d}$ and
$\boldsymbol{h}\in\mathbb{R}^{d}$ are given. Then the mixed difference of
order $\boldsymbol{k}$ with a vector step $\boldsymbol{h}$ is defined by the
equality

$\Delta^{\boldsymbol{k}}_{\boldsymbol{h}}f(\boldsymbol{x})=\Delta^{k_{1}}_{h_{1},1}\,\Delta^{k_{2}}_{h_{2},2}\cdots\,\Delta^{k_{d}}_{h_{d},d}\,f(\boldsymbol{x}).$

The spaces $B^{\boldsymbol{r}}_{p,\theta}(\mathbb{T}^{d})$, $1\leq
p,\theta\leq\infty$, where $\boldsymbol{r}\in\mathbb{R}^{d}$ is a given vector
with the elements $r_{j}>0$, $j=1,\dots,d$, are defined as follows:

$B^{\boldsymbol{r}}_{p,\theta}:=B^{\boldsymbol{r}}_{p,\theta}(\mathbb{T}^{d}):=\\{f\in
L^{0}_{p}\colon\ \|f\|_{B^{\boldsymbol{r}}_{p,\theta}}<\infty\\},$

and the respective norm is specified by

(2.1) $\displaystyle\|f\|_{B^{\boldsymbol{r}}_{p,\theta}}$
$\displaystyle:=\Bigg{(}\int_{\mathbb{T}^{d}}\|\Delta^{\boldsymbol{k}}_{\boldsymbol{h}}f\|^{\theta}_{p}\,\prod\limits^{d}_{j=1}\frac{dh_{j}}{h^{1+r_{j}{\theta}}_{j}}\Bigg{)}^{\frac{1}{\theta}},\qquad
1\leq\theta<\infty,$ (2.2) $\displaystyle\|f\|_{H^{\boldsymbol{r}}_{p}}$
$\displaystyle\equiv\|f\|_{B^{\boldsymbol{r}}_{p,\infty}}:=\sup\limits_{\boldsymbol{h}}\|\Delta^{\boldsymbol{k}}_{\boldsymbol{h}}f\|_{p}\prod\limits^{d}_{j=1}h^{-r_{j}}_{j}.$

Here we assume that the components of the vectors $\boldsymbol{k}$ and
$\boldsymbol{r}$ satisfy the conditions $k_{j}>r_{j}$, $1,\dots,d$.

In this form, the definition of the spaces $B^{\boldsymbol{r}}_{p,\theta}$,
$1\leq\theta<\infty$, was given by V.N. Temlyakov [42] and, for
$H^{\boldsymbol{r}}_{p}\equiv B^{\boldsymbol{r}}_{p,\infty}$, by S.M.
Nikol’skii and P.I. Lizorkin [20]. These spaces belong to the scale of spaces
of mixed smoothness, introduced by S.M. Nikol’skii [23] and T.I. Amanov [1].
In addition, they are generalizations of the well-known isotropic Besov spaces
[4], and the Nikol’skii spaces [22] for the case $\theta=\infty$.

In what follows, we introduce an equivalent norm representation for the spaces
$B^{\boldsymbol{r}}_{p,\theta}$, which is more convenient for calculations.

For all vectors ${\boldsymbol{s}}\in\mathbb{N}^{d}$, we set

(2.3) $\rho(\boldsymbol{s}):=\\{\boldsymbol{k}\in\mathbb{Z}^{d}\colon\
2^{s_{j}-1}\leq|k_{j}|<2^{s_{j}},\,j=1,\dots,d\\}.$

Further, for $f\in L^{0}_{p}$, let

(2.4)
$\delta_{\boldsymbol{s}}(f):=\delta_{\boldsymbol{s}}(f,\boldsymbol{x})=\sum\limits_{\boldsymbol{k}\in\rho({\boldsymbol{s}})}\,\widehat{f}(\boldsymbol{k})e^{i(\boldsymbol{k},\boldsymbol{x})},$

where
$\widehat{f}(\boldsymbol{k})=\int_{\mathbb{T}^{d}}f({\boldsymbol{t}})e^{-i(\boldsymbol{k},{\boldsymbol{t}})}d{\boldsymbol{t}}$
are the Fourier coefficients of the function $f$.

First, let $1<p<\infty$. Then we can define the norm of the spaces
$B^{\boldsymbol{r}}_{p,\theta}$, $\boldsymbol{r}\in\mathbb{R}^{d}$, $r_{j}>0$,
$j=1,\dots,d$, as follows (see, e.g., [20]):

(2.5) $\displaystyle\|f\|_{B^{\boldsymbol{r}}_{p,\theta}}$
$\displaystyle\asymp\Bigg{(}\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|\delta_{\boldsymbol{s}}(f)\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}},\quad
1\leq\theta<\infty,$ (2.6) $\displaystyle\|f\|_{H^{\boldsymbol{r}}_{p}}$
$\displaystyle\equiv\|f\|_{B^{\boldsymbol{r}}_{p,\infty}}\asymp\sup\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}2^{({\boldsymbol{s}},{\boldsymbol{r}})}\|\delta_{\boldsymbol{s}}(f)\|_{p}.$

Here and below, for positive quantities $a$ and $b$, the notation $a\asymp b$
means that there exist positive constants $C_{1}$ and $C_{2}$, that do not
depend on an essential parameter in the values of $a,b$, and such that
$C_{1}a\leq b$ (in this case, we write $a\ll b$) and $C_{2}a\geq b$ (denoted
by $a\gg b$).

All constants $C_{i},\,i=1,2,\ldots,$ in this paper may depend only on the
parameters contained in the definition of the function class, the metric in
which we estimate the approximation error, and the dimension $d$ of the space
$\mathbb{R}^{d}$. In some cases, we will specify these parameters. In the
remaining cases, they are understandable from the context.

Note that after the corresponding modification of the “blocks”
$\delta_{\boldsymbol{s}}(f)$, the given above norm representation for the
spaces $B^{\boldsymbol{r}}_{p,\theta}$ can also be generalized to the extreme
values $p=1$ and $p=\infty$ (see, e.g., [20, Remark 2.1]).

Let $V_{l}(t)$, $t\in\mathbb{R}$, $l\in\mathbb{N}$, denotes the de la Vallée-
Poussin kernel of the form

$V_{l}(t)=1+2\sum\limits^{l}_{k=1}\cos
kt+2\sum\limits^{2l-1}_{k=l+1}\Big{(}1-\frac{k-l}{l}\Big{)}\cos kt\,,$

where for $l=1$ we assume that the third term equals to zero.

We associate each vector ${\boldsymbol{s}}\in\mathbb{N}^{d}$ with the
polynomial

$A_{\boldsymbol{s}}(\boldsymbol{x})=\prod\limits^{d}_{j=1}(V_{2^{s_{j}}}(x_{j})-V_{2^{s_{j}-1}}(x_{j})),$

and for $f\in L^{0}_{p}$, $1\leq p\leq\infty$, set

$A_{\boldsymbol{s}}(f):=A_{\boldsymbol{s}}(f,\boldsymbol{x}):=(f\ast
A_{\boldsymbol{s}})(\boldsymbol{x}),$

where “$\ast$” denotes the operation of convolution. Thus, for $1\leq
p\leq\infty$, the norm of the spaces $B^{\boldsymbol{r}}_{p,\theta}$,
$\boldsymbol{r}\in\mathbb{R}^{d}$, $r_{j}>0$, $j=1,\dots,d$, can be defined as
follows:

(2.7) $\displaystyle\|f\|_{B^{\boldsymbol{r}}_{p,\theta}}$
$\displaystyle\asymp\Bigg{(}\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\,\|A_{\boldsymbol{s}}(f)\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}},\quad
1\leq\theta<\infty,$ (2.8) $\displaystyle\|f\|_{H^{\boldsymbol{r}}_{p}}$
$\displaystyle\equiv\|f\|_{B^{\boldsymbol{r}}_{p,\infty}}\asymp\sup\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\,2^{({\boldsymbol{s}},{\boldsymbol{r}})}\|A_{\boldsymbol{s}}(f)\|_{p}.$

In what follows, we will keep the notation $B^{\boldsymbol{r}}_{p,\theta}$
also for the respective classes (the unit balls in the spaces
$B^{\boldsymbol{r}}_{p,\theta}$). Further, for the classes, we will use the
respective definition of the norm from (2.5), (2.6) or (2.7), (2.8) depending
on the value of the parameter $p$. This should not create any confusion, since
we study the asymptotic characteristics of the classes of functions. These
definitions of the norm, as noted, are equivalent to (2.1) and (2.2).

For forerunners in the investigation of different approximation
characteristics of the Nikol’skii and Nikol’skii-Besov classes of periodic
functions we refer to [8, 30, 42, 45, 48], and the references therein.

Note, that with a growth of the parameter $\theta$ the classes
$B^{\boldsymbol{r}}_{p,\theta}$ are expanding, i.e.,

$B^{\boldsymbol{r}}_{p,1}\subset B^{\boldsymbol{r}}_{p,\theta_{1}}\subset
B^{\boldsymbol{r}}_{p,\theta_{2}}\subset B^{\boldsymbol{r}}_{p,\infty}\equiv
H^{\boldsymbol{r}}_{p},\quad 1\leq\theta_{1}\leq\theta_{2}\leq\infty.$

Further we assume that coordinates of the vector
${\boldsymbol{r}}\in\mathbb{R}^{d}$, as the parameter of the defined classes,
are ordered such that $0<r_{1}=r_{2}=\dots=r_{\nu}<r_{\nu+1}\leq\dots\leq
r_{d}$, and also that ${\boldsymbol{{\boldsymbol{\gamma}}}}\in\mathbb{R}^{d}$
is a vector with the coordinates ${\gamma}_{j}=r_{j}/r_{1}$, $j=1,\dots,d$.
Besides, ${\boldsymbol{{\boldsymbol{\gamma}}}}^{\prime}\in\mathbb{R}^{d}$,
where $\gamma^{\prime}_{j}=\gamma_{j}=1$ if $j=1,\dots,\nu$ and
$1<\gamma^{\prime}_{j}<\gamma_{j}$ if $j=\nu+1,\dots,d$.

In what follows, we define the norm $\|\cdot\|_{B_{q,1}}$ in the subspaces
$B_{q,1}\subset L^{0}_{q}$, $1\leq q\leq\infty$.

For trigonometric polynomials $t$ with respect to the trigonometric system
$\\{e^{i(\boldsymbol{k},\boldsymbol{x})}\\}_{\boldsymbol{k}\in\mathbb{Z}^{d}}$,
the norm
$\|t\|_{B_{q,1}}:=\sum_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|A_{\boldsymbol{s}}(t)\|_{q}$,
$1\leq q\leq\infty$ (the sum contains a finite number of terms).

Similarly we define the norm for functions $f\in L^{0}_{q}$, such that the
series $\sum_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|A_{\boldsymbol{s}}(f)\|_{q}$
is convergent:

(2.9)
$\|f\|_{B_{q,1}}:=\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|A_{\boldsymbol{s}}(f)\|_{q},\quad
1\leq q\leq\infty.$

Note, that in the case $1<q<\infty$ it holds

(2.10)
$\|f\|_{B_{q,1}}\asymp\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|\delta_{\boldsymbol{s}}(f)\|_{q}.$

For $f\in B_{q,1}$, $1\leq q\leq\infty$, the following relations hold:

$\|f\|_{q}\ll\|f\|_{B_{q,1}};\qquad\|f\|_{B_{1,1}}\ll\|f\|_{B_{q,1}}\ll\|f\|_{B_{\infty,1}}.$

##  3\. Approximation by the step hyperbolic Fourier sums and
best approximation

First, we give definitions of the approximation characteristics, which are
investigated in this part of the paper.

For $n\in\mathbb{N}$, ${\boldsymbol{s}}\in\mathbb{N}^{d}$ and
${\boldsymbol{\gamma}}\in\mathbb{R}^{d}$, $\gamma_{j}>0$, $j=1,\dots,d$, we
put

$Q^{\boldsymbol{\gamma}}_{n}:=\bigcup\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})<n}\,\rho({\boldsymbol{s}}),$

where $\rho({\boldsymbol{s}})$ is defined by (2.3). The set
$Q^{{{\boldsymbol{\gamma}}}}_{n}$ is called the step hyperbolic cross. In the
case ${{\boldsymbol{\gamma}}}=(1,\ldots,1):=\boldsymbol{1}\in\mathbb{N}^{d}$,
we use the notation $Q^{\boldsymbol{1}}_{n}$.

Further, we consider the set of trigonometric polynomials

$T(Q^{{\boldsymbol{\gamma}}}_{n}):=\Big{\\{}t\colon\
t(\boldsymbol{x})=\sum\limits_{\boldsymbol{k}\in
Q^{{\boldsymbol{\gamma}}}_{n}}c_{\boldsymbol{k}}e^{i(\boldsymbol{k},\boldsymbol{x})},\,c_{\boldsymbol{k}}\in\mathbb{C},\
\boldsymbol{x}\in\mathbb{R}^{d}\Big{\\}}$

and for $f\in L^{0}_{1}$ define

$S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f):=S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f,\boldsymbol{x}):=\sum\limits_{\boldsymbol{k}\in
Q^{{\boldsymbol{\gamma}}}_{n}}\widehat{f}(\boldsymbol{k})\,e^{i(\boldsymbol{k},\boldsymbol{x})},\quad\boldsymbol{x}\in\mathbb{R}^{d},$

where $\widehat{f}(\boldsymbol{k})$ are the Fourier coefficients of the
function $f$.

The polynomials $S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)$ are called the step
hyperbolic Fourier sums of functions $f$. According to (2.4), they can be
written in the form

$S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})<n}\delta_{\boldsymbol{s}}(f).$

In connection to the above defined sets of polynomials, we will consider the
following approximation characteristics.

Let $\mathscr{X}$ be a $d$-dimensional, $d\geq 1$, functional space equipped
with the norm $\|\cdot\|_{\mathscr{X}}$. Then, for $f\in\mathscr{X}$, by

$E_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{\mathscr{X}}:=\inf\limits_{t\in
T(Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n})}\|f-t\|_{\mathscr{X}}$

we define a quantity of the best approximation of the function $f$ by
polynomials from the set $T(Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n})$.

Respectively, for the functional class $F\subset\mathscr{X}$, we set

(3.1)
$E_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(F)_{\mathscr{X}}:=\sup\limits_{f\in
F}E_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{\mathscr{X}}.$

Along with the quantities (3.1), we will investigate the approximation
characteristics

(3.2)
$\mathscr{E}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(F)_{\mathscr{X}}:=\sup\limits_{f\in
F}\,\|f-S_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)\|_{\mathscr{X}}.$

An investigation of the approximation properties of (3.1) and (3.2) on the
Sobolev classes $W^{r}_{p,\alpha}$ and Nikol’skii-Besov classes
$B^{\boldsymbol{r}}_{p,\theta}$ in the case $\mathscr{X}=L_{q}$ has a reach
history. We refer to the monographs [8, 30, 42, 45, 48].

The aim of this part of the paper is to get estimates for the quantities (3.1)
and (3.2) in the case $F=B^{\boldsymbol{r}}_{p,\theta}$ and
$\mathscr{X}=B_{q,1}$.

First, we note the existing connection between the quantities
$E_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{B_{q,1}}$ and
$\mathscr{E}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{B_{q,1}}$.

From the one hand, immediately from the definitions (3.1) and (3.2), we obtain
that for the functions $f\in B_{q,1}$ it holds

(3.3)
$E_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{B_{q,1}}\leq\mathscr{E}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{B_{q,1}},\qquad
1\leq q\leq\infty.$

Further we show, that in the case $1<q<\infty$ it also holds that

$\mathscr{E}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{B_{q,1}}\ll
E_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)_{B_{q,1}}.$

Let $\textbf{S}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}$ denotes the
Fourier operator that puts into a correspondence to the function $f\in
B_{q,1}$, $1<q<\infty$, its step hyperbolic Fourier sum
$S_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)$, i.e.,
$\textbf{S}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}f=S_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}(f)$.

Let us show the boundedness of
$\|\textbf{S}_{Q^{{\boldsymbol{{\boldsymbol{\gamma}}}}}_{n}}\|_{B_{q,1}\rightarrow
B_{q,1}}$, $1<q<\infty$.

Indeed, due to (2.10), we have

$\displaystyle\|\textbf{S}_{Q^{{\boldsymbol{\gamma}}}_{n}}\|_{B_{q,1}\rightarrow
B_{q,1}}$ $\displaystyle=\sup\limits_{\|f\|_{B_{q,1}}\leq
1}\|S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)\|_{B_{q,1}}=\sup\limits_{\|f\|_{B_{q,1}}\leq
1}\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|\delta_{\boldsymbol{s}}(S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f))\|_{q}$
$\displaystyle=\sup\limits_{\|f\|_{B_{q,1}}\leq
1}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})<n}\|\delta_{\boldsymbol{s}}(f)\|_{q}\leq\sup\limits_{\|f\|_{B_{q,1}}\leq
1}\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|\delta_{\boldsymbol{s}}(f)\|_{q}$
(3.4) $\displaystyle\asymp\sup\limits_{\|f\|_{B_{q,1}}\leq
1}\|f\|_{B_{q,1}}\leq 1.$

Further, let $t^{\ast}\in T(Q^{{\boldsymbol{\gamma}}}_{n})$ be a polynomial of
the best approximation of the function $f\in B_{q,1}$ in the space $B_{q,1}$.
Then, taking into account the fact that
$S_{Q^{{\boldsymbol{\gamma}}}_{n}}(t^{\ast})=t^{\ast}$ and using (3.4), we can
write

$\displaystyle\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}$
$\displaystyle=\|f-S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)\|_{B_{q,1}}=\|f-t^{\ast}+t^{\ast}-S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)\|_{B_{q,1}}$
$\displaystyle\leq\|f-t^{\ast}\|_{B_{q,1}}+\|S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)-t^{\ast}\|_{B_{q,1}}$
$\displaystyle=\|f-t^{\ast}\|_{B_{q,1}}+\|S_{Q^{{\boldsymbol{\gamma}}}_{n}}(f-t^{\ast})\|_{B_{q,1}}$
$\displaystyle\leq
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}+\|\textbf{S}_{Q^{{\boldsymbol{\gamma}}}_{n}}\|_{B_{q,1}\rightarrow
B_{q,1}}\|f-t^{\ast}\|_{B_{q,1}}$ (3.5) $\displaystyle\leq
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}+C_{3}(q)\cdot
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}=(1+C_{3}(q))\cdot
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}.$

Combining (3.3) and (3.5), we get

$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}\asymp\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}},\quad
1<q<\infty.$

Let us formulate an auxiliary statement that we will use multiple times in
what follows.

Lemma A. [42, Lemmas B, p. 11] Let ${\boldsymbol{s}}\in\mathbb{N}^{d}$,
${\boldsymbol{\gamma}}\in\mathbb{R}^{d}$, $\gamma_{j}>0$, $j=1,\dots,d$. Then
for $\alpha>0$, the following estimate holds:

$\sum\limits_{(s,{\boldsymbol{\gamma}})\geq
l}2^{-\alpha(s,{\boldsymbol{\gamma}})}\asymp 2^{-\alpha l}l^{d-1}.$

If ${\boldsymbol{\gamma}}^{\prime}\in\mathbb{R}^{d}$ is such that
${\gamma}_{j}={\gamma}_{j}^{\prime}=1$ for $j=1,\dots,\nu$ and
$1<\gamma_{j}^{\prime}<\gamma_{j}$ for $j=\nu+1,\dots,d$. Then for $\alpha>0$,
it holds:

$\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
l}2^{-\alpha({\boldsymbol{s}},{\boldsymbol{\gamma}})}\asymp 2^{-\alpha
l}l^{\nu-1}.$

Theorem A. Let

$t(\boldsymbol{x})=\sum\limits_{|k_{j}|\leq
n_{j}}c_{\boldsymbol{k}}e^{i(\boldsymbol{k},\boldsymbol{x})},$

where $\boldsymbol{x}\in\mathbb{R}^{d}$, $\boldsymbol{k}\in\mathbb{Z}^{d}$,
$\boldsymbol{n}\in\mathbb{N}^{d}$, $c_{\boldsymbol{k}}\in\mathbb{C}$. Then for
$1\leq p<q\leq\infty$ the following inequality holds:

(3.6) $\|t\|_{q}\leq
2^{d}\prod\limits^{d}_{j=1}n_{j}^{\frac{1}{p}-\frac{1}{q}}\|t\|_{p}.$

Inequality (3.6) was obtained by S.M. Nikol’skii [22] and is referred to as
the inequality for different metrics.

Further we move to formulation and proving the results.

First, we obtain the exact order estimates for the quantities
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}$
and
$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}$
in the case $1<p<q<\infty$.

######  Theorem 3.1.

Let $d\geq 2$, $1<p<q<\infty$, $1\leq\theta\leq\ \infty$. Then for
$r_{1}>1/p-1/q$ the following estimates hold

(3.7)
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}\asymp
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}\asymp
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}n^{(\nu-1)(1-\frac{1}{\theta})}.$

###### Proof.

To get the upper estimates in (3.7), it is sufficient, due to (3.3), to
respectively estimate the quantity
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}$.

Let $f$ be a function from the class $B^{\boldsymbol{r}}_{p,\theta}$. Then,
from the equivalent representation (2.10) of the norm in the space $B_{q,1}$,
we have

$\displaystyle\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(f)_{B_{q,1}}$
$\displaystyle=\Big{\|}f-\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})<n}\delta_{\boldsymbol{s}}(f)\Big{\|}_{B_{q,1}}=\Big{\|}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}\delta_{\boldsymbol{s}}(f)\Big{\|}_{B_{q,1}}$ (3.8)
$\displaystyle\asymp\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\Bigg{\|}\delta_{\boldsymbol{s}}\Bigg{(}\mathop{\sum_{{\boldsymbol{s}}^{\prime}\in\mathbb{N}^{d}}}\limits_{({\boldsymbol{s}}^{\prime},{\boldsymbol{\gamma}})\geq
n}\delta_{{\boldsymbol{s}}^{\prime}}(f)\Bigg{)}\Bigg{\|}_{q}\leq\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}\|\delta_{\boldsymbol{s}}(f)\|_{q}=:J_{1}.$

To further estimate the quantity $J_{1}$, let us distinguish three cases
depending on the value of the parameter $\theta$.

Case $\theta=1$. Then, taking into account Theorem A and the fact that
$({\boldsymbol{s}},{\boldsymbol{1}})\leq({\boldsymbol{s}},{\boldsymbol{\gamma}})$,
we can write

$\displaystyle J_{1}$
$\displaystyle=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})r_{1}}2^{({\boldsymbol{s}},{\boldsymbol{1}})(\frac{1}{p}-\frac{1}{q})}2^{({\boldsymbol{s}},{\boldsymbol{r}})}\|\delta_{\boldsymbol{s}}(f)\|_{p}$
$\displaystyle\leq\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})(r_{1}-\frac{1}{p}+\frac{1}{q})}2^{({\boldsymbol{s}},{\boldsymbol{r}})}\|\delta_{\boldsymbol{s}}(f)\|_{p}$
$\displaystyle\leq
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}\sum\limits_{(\boldsymbol{s},{\boldsymbol{\gamma}})\geq
n}2^{(\boldsymbol{s},\boldsymbol{r})}\|\delta_{\boldsymbol{s}}(f)\|_{p}$
$\displaystyle\leq
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}\|f\|_{B^{\boldsymbol{r}}_{p,1}}\leq
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}.$

Case $1<\theta<\infty$. Using the Hölder’s inequality with the power $\theta$
to $J_{1}$, we get

$\displaystyle J_{1}$
$\displaystyle=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-(({\boldsymbol{s}},{\boldsymbol{r}})-({\boldsymbol{s}},{\boldsymbol{1}})(\frac{1}{p}-\frac{1}{q}))}2^{({\boldsymbol{s}},{\boldsymbol{r}})}\|\delta_{\boldsymbol{s}}(f)\|_{p}$
$\displaystyle\leq\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|\delta_{\boldsymbol{s}}(f)\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-(({\boldsymbol{s}},{\boldsymbol{r}})-({\boldsymbol{s}},{\boldsymbol{1}})(\frac{1}{p}-\frac{1}{q}))\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}}$
(3.9)
$\displaystyle\ll\|f\|_{B^{\boldsymbol{r}}_{p,\theta}}\,\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{r}}-\frac{1}{p}+\frac{1}{q})\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}}\leq\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},\widetilde{{\boldsymbol{\gamma}}})(r_{1}-\frac{1}{p}+\frac{1}{q})\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}},$

where ${\boldsymbol{r}}-1/p+1/q$ is a vector with the coordinates
$r_{j}-1/p+1/q$, $j=1,\dots,d$, and, respectively,
$\widetilde{{\boldsymbol{\gamma}}}\in\mathbb{R}^{d}$ is such that
$\widetilde{\gamma}_{j}=(r_{j}-1/p+1/q)/(r_{1}-1/p+1/q)$, $j=1,\dots,d$.

One can easily check that $\widetilde{\gamma}_{j}={\gamma}_{j}=1$ for
$j=1,\dots,\nu$ and $1<{\gamma}_{j}<\widetilde{{\gamma}}_{j}$ for
$j=\nu+1,\dots,d$. From this, using Lemma A for the last sum in (3.9), we
obtain

$J_{1}\ll
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}n^{(\nu-1)(1-\frac{1}{\theta})}.$

Case $\theta=\infty$. Taking into account, that for $f\in
B^{\boldsymbol{r}}_{p,\infty}$, $1<p<\infty$, it holds (see (2.6))

(3.10) $\|\delta_{\boldsymbol{s}}(f)\|_{p}\ll
2^{-({\boldsymbol{s}},{\boldsymbol{r}})},\quad{\boldsymbol{s}}\in\mathbb{N}^{d},$

using Lemma A, we have that

$\displaystyle J_{1}$
$\displaystyle\ll\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-(({\boldsymbol{s}},{\boldsymbol{r}})-({\boldsymbol{s}},{\boldsymbol{1}})(\frac{1}{p}-\frac{1}{q}))}=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{r}}-\frac{1}{p}+\frac{1}{q})}$ (3.11)
$\displaystyle=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},\widetilde{{\boldsymbol{\gamma}}})(r_{1}-\frac{1}{p}+\frac{1}{q})}\ll
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}n^{\nu-1}.$

Combining (3.8) – (3.11), we obtain the upper estimate for the quantity
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}$,
and hence for
$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}$.

It is sufficient to get the lower estimate in (3.7) for the case $\nu=d$.
Further we construct a function that realizes the obtained upper order
estimate.

Let

$g(\boldsymbol{x})=C_{4}\,2^{-n(r_{1}+1-\frac{1}{p})}n^{-{\frac{d-1}{\theta}}}d_{n}(\boldsymbol{x}),\quad
C_{4}>0,$

where

$d_{n}(\boldsymbol{x})=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}\,\sum\limits_{\boldsymbol{k}\in\rho({\boldsymbol{s}})}e^{i(\boldsymbol{k},\boldsymbol{x})}.$

Let us first show that for certain choice of the constant $C_{4}$, $g\in
B^{\boldsymbol{r}}_{p,\theta}$,
${\boldsymbol{r}}=(r_{1},\dots,r_{1})\in\mathbb{R}^{d}$, $r_{1}>0$,
$1<p<\infty$, $1\leq\theta\leq\infty$.

Let first $1\leq\theta<\infty$. Then we can write

$\displaystyle\|g\|_{B^{\boldsymbol{r}}_{p,\theta}}$
$\displaystyle\asymp\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|\delta_{\boldsymbol{s}}(g)\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}}$
$\displaystyle\asymp
2^{-n(r_{1}+1-\frac{1}{p})}n^{-{\frac{d-1}{\theta}}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},\boldsymbol{1})=n}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|\delta_{\boldsymbol{s}}(d_{n})\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}}$
(3.12)
$\displaystyle=2^{-n(1-\frac{1}{p})}n^{-{\frac{d-1}{\theta}}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}\|\delta_{\boldsymbol{s}}(d_{n})\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}}.$

Further we use the known relation

$\Big{\|}\sum\limits^{m}_{k=-m}e^{ikx}\Big{\|}_{p}\asymp
m^{1-\frac{1}{p}},\quad 1<p<\infty,\ x\in\mathbb{R}$

(see, e.g., [45, Ch. 1, Paragraph 1]), and get

(3.13) $\|\delta_{\boldsymbol{s}}(d_{n})\|_{p}\asymp
2^{({\boldsymbol{s}},{\boldsymbol{1}})(1-\frac{1}{p})}.$

Substituting (3.13) into (3.12), we continue the estimation of
$\|g\|_{B^{\boldsymbol{r}}_{p,\theta}}$ as follows

$\displaystyle\|g\|_{B^{\boldsymbol{r}}_{p,\theta}}$ $\displaystyle\asymp
2^{-n(1-\frac{1}{p})}n^{-{\frac{d-1}{\theta}}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}2^{({\boldsymbol{s}},{\boldsymbol{1}})(1-\frac{1}{p})\theta}\Bigg{)}^{\frac{1}{\theta}}$
$\displaystyle=n^{-{\frac{d-1}{\theta}}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}1\Bigg{)}^{\frac{1}{\theta}}\asymp
n^{-{\frac{d-1}{\theta}}}n^{\frac{d-1}{\theta}}=1.$

Let further $\theta=\infty$. Then the function $g$ takes the form

$g(\boldsymbol{x})=C_{4}\,2^{-n(r_{1}+1-\frac{1}{p})}d_{n}(\boldsymbol{x})$

and therefore, in view of (2.5) and (3.13), we get

$\|g\|_{B^{\boldsymbol{r}}_{p,\infty}}\asymp\sup\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}2^{({\boldsymbol{s}},{\boldsymbol{r}})}\|\delta_{\boldsymbol{s}}(g)\|_{p}\asymp\sup\limits_{{\boldsymbol{s}}:({\boldsymbol{s}},{\boldsymbol{1}})=n}\,2^{({\boldsymbol{s}},{\boldsymbol{r}})}2^{-n(r_{1}+1-\frac{1}{p})}\|\delta_{\boldsymbol{s}}(d_{n})\|_{p}\asymp
1.$

We make a conclusion that for an appropriate choice of the constant $C_{4}>0$,
the function $g\in B^{\boldsymbol{r}}_{p,\theta}$.

Further, taking into account that $S_{Q^{\boldsymbol{1}}_{n}}(g)=0$ and the
relation (3.13), we can write

$\displaystyle E_{Q^{\boldsymbol{1}}_{n}}(g)_{B_{q,1}}$
$\displaystyle\asymp\mathscr{E}_{Q^{\boldsymbol{1}}_{n}}(g)_{B_{q,1}}=\|g\|_{B_{q,1}}\asymp
2^{-n(r_{1}+1-\frac{1}{p})}n^{-{\frac{d-1}{\theta}}}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}\|\delta_{\boldsymbol{s}}(d_{n})\|_{q}$
$\displaystyle\asymp
2^{-n(r_{1}+1-\frac{1}{p})}n^{-{\frac{d-1}{\theta}}}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{1}})=n}2^{({\boldsymbol{s}},{\boldsymbol{1}})(1-\frac{1}{q})}\asymp
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}n^{(d-1)(1-\frac{1}{\theta})}.$

Theorem 3.1 is proved ∎

In what follows, we separately consider the case $p=q$, since in this
situation the orders of the corresponding approximation quantities for
$Q^{{\boldsymbol{\gamma}}}_{n}$ and $Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}$
with $\gamma^{\prime}_{j}=\gamma_{j}=1$ if $j=1,\dots,\nu$ and
$1<\gamma^{\prime}_{j}<\gamma_{j}$ if $j=\nu+1,\dots,d$, differ.

######  Theorem 3.2.

Let $d\geq 2$, $1<p<\infty$, $1\leq\theta\leq\infty$. Then for $r_{1}>0$ it
holds

$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}\asymp
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

###### Proof.

We will argue similarly to that in Theorem 3.1. For $f\in
B^{\boldsymbol{r}}_{p,\theta}$, we obtain

$\displaystyle\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(f)_{B_{p,1}}\leq\sum\limits_{(s,{\boldsymbol{\gamma}}^{\prime})\geq
n}\|\delta_{s}(f)\|_{p}=:J_{2}.$

Further, as above, we distinguish three cases.

For the case $\theta=1$, we note that
$2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})r_{1}}\leq
2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})r_{1}}$.

In the case $1<\theta<\infty$, by Lemma A, in view of
${\boldsymbol{\gamma}}={\boldsymbol{r}}/r_{1}$, we obtain

$\displaystyle J_{2}$
$\displaystyle\ll\|f\|_{B^{\boldsymbol{r}}_{p,\theta}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{r}})\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}}\leq\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})r_{1}\,\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}}$
$\displaystyle\ll 2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

Case $\theta=\infty$. In view of (3.10), Lemma A yields

$J_{2}\ll\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{r}})}\ll 2^{-nr_{1}}n^{\nu-1}.$

For the respective lower estimate, it is sufficient to consider the function
$g\in B^{\boldsymbol{r}}_{p,\theta}$ from the proof of Theorem 3.1.

Theorem 3.2 is proved. ∎

######  Remark 3.3.

Under the conditions of Theorem 3.2, for the quantities
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}$
and
$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}$
the following relations hold

(3.14)
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}\asymp
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}\asymp
2^{-nr_{1}}n^{(d-1)(1-\frac{1}{\theta})}.$

This is a corollary from (see Lemma A)

$\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{r}})}=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}})\geq
n}2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})r_{1}}\asymp
2^{-nr_{1}}n^{d-1}.$

Further we comment on the obtained in Theorems 3.1 and 3.2 results.

First, let us note that the corresponding statements, where the error norm is
measured in $L_{q}$, are known. Let us formulate them for convenience.

Theorem B. Let $d\geq 2$. Then for $r_{1}>0$ it holds

$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{p}\asymp
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{p}\asymp\left\\{\begin{array}[]{ll}2^{-nr_{1}}n^{(\nu-1)(\frac{1}{p}-\frac{1}{\theta})},&1<p\leq
2,\ p<\theta\leq\infty,\\\
2^{-nr_{1}}n^{(\nu-1)(\frac{1}{2}-\frac{1}{\theta})},&2<p<\infty,\
2<\theta\leq\infty.\end{array}\right.$

In the cases $1<p\leq 2$ and $1\leq\theta\leq p$, or $2<p<\infty$ and
$1\leq\theta\leq 2$ it holds

$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{p}\asymp
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{p}\asymp
2^{-nr_{1}}.$

In the case $\theta=\infty$, Theorem B was proved in [5] for $p=2$ and in [24]
for $1<p<\infty$. In the case $1\leq\theta<\infty$, the corresponding
estimates were obtained in [25].

Theorem C. Let $d\geq 2$, $1<p<q<\infty$, $1\leq\theta\leq\infty$. Then for
$r_{1}>1/p-1/q$ the following relations hold

$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{q}\asymp
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{q}\asymp
2^{-n(r_{1}-\frac{1}{p}+\frac{1}{q})}n^{(\nu-1)(\frac{1}{q}-\frac{1}{\theta})_{+}},$

where $a_{+}=\max\\{a,0\\}$.

In the case $\theta=\infty$, Theorem C was proved in [41] for $1<p<q\leq 2$
and in [42] for $1<p<q<\infty$, $q>2$, respectively. In the case
$1\leq\theta<\infty$, the corresponding estimates were obtained in [25].

######  Remark 3.4.

Comparing the results of Theorems 3.1 and 3.2 (see also (3.14)) and Theorems B
and C, we make the following conclusion. Only in the cases $\theta=1$ or
$\nu=1$ the corresponding approximation characteristics of the classes
$B^{\boldsymbol{r}}_{p,\theta}$ in the spaces $B_{q,1}$ and $L_{q}$, $1<p\leq
q<\infty$, coincide in order and, if additionally $p=q$, they do not depend on
the parameter $p$. In all other cases, their orders differ.

Returning to Theorem 3.2, we note that it does not cover the limit cases
$p\in\\{1,\infty\\}$.

The following statement concerns these limit cases, but only for the quantity
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}$.

######  Theorem 3.5.

Let $d\geq 2$, $p\in\\{1,\infty\\}$, $1\leq\theta\leq\infty$. Then for
$r_{1}>0$ it holds

(3.15)
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

###### Proof.

We begin with the upper estimate. As an approximation aggregate for a function
$f\in B^{\boldsymbol{r}}_{p,\theta}$, we take the polynomial

(3.16)
$t_{n}:=t_{n}(f,\boldsymbol{x})=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})<n-({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}A_{\boldsymbol{s}}(f,\boldsymbol{x}),\quad\boldsymbol{x}\in\mathbb{R}^{d}.$

One can easily check that $t_{n}\in
T(Q^{{\boldsymbol{\gamma}}^{\prime}}_{n})$.

Then, by the definition (2.9) of the norm in the space $B_{p,1}$,
$p\in\\{1,\infty\\}$, we can write

$\displaystyle E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(f)_{B_{p,1}}$
$\displaystyle\leq\|f-t_{n}\|_{B_{p,1}}=\Big{\|}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}\,A_{\boldsymbol{s}}(f)\Big{\|}_{B_{p,1}}$
$\displaystyle=\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\,\Bigg{\|}A_{\boldsymbol{s}}\ast\,\mathop{\sum_{{\boldsymbol{s}}^{\prime}\in\mathbb{N}^{d}}}\limits_{({\boldsymbol{s}}^{\prime},{\boldsymbol{\gamma}}^{\prime})\geq
n-({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}\,A_{{\boldsymbol{s}}^{\prime}}(f)\Bigg{\|}_{p}$
$\displaystyle\leq\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-2({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}\,\Big{\|}A_{\boldsymbol{s}}\ast\sum\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}\,A_{{\boldsymbol{s}}^{\prime}}(f)\Big{\|}_{p}$ (3.17)
$\displaystyle\leq\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-2({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}\,\|A_{\boldsymbol{s}}\|_{1}\,\Big{\|}\sum\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}\,A_{{\boldsymbol{s}}^{\prime}}(f)\Big{\|}_{p}.$

Further we use the fact that, in view of the relation $\|V_{2^{s}}\|_{1}\leq
C_{5}$, $C_{5}>0$ (see, e.g., [45, Ch. 1, Paragraph 1]), it holds

(3.18)
$\|A_{\boldsymbol{s}}\|_{1}=\Big{\|}\prod_{j=1}^{d}(V_{2^{s_{j}}}-V_{2^{s_{j}-1}})\Big{\|}_{1}\leq\Big{\|}\prod_{j=1}^{d}V_{2^{s_{j}}}\Big{\|}_{1}+\Big{\|}\prod_{j=1}^{d}V_{2^{s_{j}-1}}\Big{\|}_{1}\leq
C_{6},\quad C_{6}>0.$

Hence, from (3.17) we obtain

$\displaystyle E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(f)_{B_{p,1}}$
$\displaystyle\ll\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-2({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}\,\sum\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}\,\|A_{{\boldsymbol{s}}^{\prime}}(f)\|_{p}\ll\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}\|A_{\boldsymbol{s}}(f)\|_{p}$
(3.19)
$\displaystyle=\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}2^{({\boldsymbol{s}},{\boldsymbol{r}})}\,\|A_{\boldsymbol{s}}(f)\|_{p}\,2^{-({\boldsymbol{s}},{\boldsymbol{r}})}.$

Let first $1\leq\theta<\infty$. Using the Hölder’s inequality with the power
$\theta$ (with an appropriate modification in the case $\theta=1$), and then
in view of (2.7), ${\boldsymbol{\gamma}}={\boldsymbol{r}}/r_{1}$ and Lemma A,
from (3.19) we get

$\displaystyle E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(f)_{B_{p,1}}$
$\displaystyle\ll\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|A_{\boldsymbol{s}}(f)\|^{\theta}_{p}\Bigg{)}^{\frac{1}{\theta}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}2^{-({\boldsymbol{s}},{\boldsymbol{r}})\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}}$
$\displaystyle\ll\|f\|_{B^{\boldsymbol{r}}_{p,\theta}}\Bigg{(}\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})r_{1}\theta^{\prime}}\Bigg{)}^{\frac{1}{\theta^{\prime}}}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

If $\theta=\infty$, then, in view of the relation
$\|A_{\boldsymbol{s}}(f)\|_{p}\ll 2^{-({\boldsymbol{s}},{\boldsymbol{r}})}$
(see (2.8)), from (3.19) for $p\in\\{1,\infty\\}$ we get

$\displaystyle
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(f)_{B_{p,1}}\ll\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},1)}\|A_{\boldsymbol{s}}(f)\|_{p}\ll\sum\limits_{({\boldsymbol{s}},{\boldsymbol{\gamma}}^{\prime})\geq
n-3({\boldsymbol{\gamma}}^{\prime},{\boldsymbol{1}})}2^{-({\boldsymbol{s}},{\boldsymbol{\gamma}})r_{1}}\asymp
2^{-nr_{1}}n^{\nu-1}.$

The upper estimate is now proved.

Concerning the lower estimate in (3.15), we note the following.

The estimate of the quantity
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{1,\theta})_{B_{1,1}}$
is a corollary from Theorem D (see below) and the relation
$\|\cdot\|_{B_{1,1}}\gg\|\cdot\|_{1}$. The lower estimate of
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{B_{\infty,1}}$
follows from the corresponding estimate of the Kolmogorov widths (see Theorem
4.2 and Remark 4.3), that will be obtained in Section 4.

Theorem 3.5 is proved. ∎

######  Remark 3.6.

One can show that under the conditions of Theorem 3.5 it holds

(3.20)
$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}\asymp
2^{-nr_{1}}n^{(d-1)(1-\frac{1}{\theta})}.$

In what follows, we comment the obtained in Theorem 3.5 result.

To compare the estimate (3.15) with the corresponding results in the space
$L_{p}$, $p\in\\{1,\infty\\}$, let us formulate the known statements.

Theorem D.  Let $d\geq 2$, $1\leq\theta\leq\infty$. Then for $r_{1}>0$ it
holds

(3.21)
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{1,\theta})_{1}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

The estimate (3.21) in the case $\theta=\infty$ was obtained in [39], and in
the case $1\leq\theta<\infty$ in [26].

Theorem E. Let $d=2$, $1\leq\theta\leq\infty$. Then for
$\boldsymbol{r}=(r_{1},r_{1})$, $r_{1}>0$, it holds

(3.22)
$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{\infty}\asymp
2^{-nr_{1}}n^{1-\frac{1}{\theta}}.$

The estimate (3.22) in the case $\theta=\infty$ was proved in [39], and in the
case $1\leq\theta<\infty$ in [29].

######  Remark 3.7.

A question concerning the orders of the quantity
$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{\infty}$
for $d>2$ remains open. In particular, for the classes
$H^{\boldsymbol{r}}_{\infty}$ see [8, Open Problem 4.7].

######  Remark 3.8.

Comparing Theorem 3.5 for $p=1$ with Theorem D we conclude that for $d\geq 2$,
$1\leq\theta\leq\infty$ and $r_{1}>0$ the following relations hold

$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{1,\theta})_{B_{1,1}}\asymp
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{1,\theta})_{1}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

Similarly, comparing Theorem 3.5 for $p=\infty$ (see also (3.20)) with Theorem
E we see that for $d=2$, $1\leq\theta\leq\infty$ and
$\boldsymbol{r}=(r_{1},r_{1})$, $r_{1}>0$, it holds

$E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{B_{\infty,1}}\asymp
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{\infty}\asymp
2^{-nr_{1}}n^{1-\frac{1}{\theta}}.$

######  Remark 3.9.

The orders of the quantities
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}$
and
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{p}$,
$p\in\\{1,\infty\\}$, $1\leq\theta\leq\infty$, $r_{1}>0$ in the case $d\geq 2$
remain unknown.

In view of (3.15), (3.20) and (3.22), we formulate the conjecture.

Conjecture 1. Let $d>2$, $1\leq\theta\leq\infty$. Then for $r_{1}>0$ it holds

$\displaystyle
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{\infty}$
$\displaystyle\asymp 2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})},$
$\displaystyle
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{\infty}$
$\displaystyle\asymp 2^{-nr_{1}}n^{(d-1)(1-\frac{1}{\theta})}.$

To conclude this part of the paper, we consider one more relation between the
parameters $p$ and $q$.

######  Theorem 3.10.

Let $d\geq 2$, $1\leq q<p\leq\infty$, $1\leq\theta\leq\infty$. Then for
$r_{1}>0$ it holds

(3.23)
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}\asymp
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

###### Proof.

The upper estimates follow from Theorem 3.2. Indeed, taking into account that
$\|\cdot\|_{B_{1,1}}\ll\|\cdot\|_{B_{q,1}}$, $1\leq q<\infty$, it is
sufficient to prove these estimates in the case $1<q<p\leq\infty$. From the
other side, due to the embeddings $B^{\boldsymbol{r}}_{\infty,\theta}\subset
B^{\boldsymbol{r}}_{p,\theta}\subset B^{\boldsymbol{r}}_{q,\theta}$, we can
restrict ourself in considering the case $1<p=q<\infty$, i.e., use the
corresponding estimates obtained in Theorem 3.2.

As for the lower estimates in (3.23), we note that they follow from the
corresponding estimates for the Kolmogorov width (see Remark 4.3 to Theorem
4.2), that will be obtained in Section 4.

Theorem 3.10 is proved. ∎

Let us further recall the known statements for the corresponding approximation
characteristics in the space $L_{q}$.

Theorem F. Let $d\geq 2$, $1<q<p\leq\infty$, $p^{\ast}=\min\\{2,p\\}$,
$r_{1}>0$. Then for $p^{\ast}<\theta\leq\infty$ it holds

(3.24)
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{q}\asymp
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{q}\asymp
2^{-nr_{1}}n^{(\nu-1)(\frac{1}{p^{\ast}}-\frac{1}{\theta})}.$

In the case $1\leq\theta\leq p^{\ast}$, it holds

$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{q}\asymp
E_{Q^{{\boldsymbol{\gamma}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{q}\asymp
2^{-nr_{1}}.$

For the classes $B^{\boldsymbol{r}}_{p,\infty}$ in the case $2\leq
q<p\leq\infty$ the corresponding upper bounds follow from the upper bounds for
the case $1<q=p<\infty$ (see Theorem B) and the lower bounds follow from [40].
In the case $1<q<2\leq p<\infty$, Theorem F was proved in [7, 11]. In the case
$1\leq q<p\leq 2$, the proof of the lower bounds required a new technique, see
[43]. For $1\leq\theta<\infty$, the corresponding statement was obtained in
[25].

Theorem G. Let $d\geq 2$, $1<p\leq\infty$, $1\leq\theta\leq\infty$. Then for
$r_{1}>0$ it holds

(3.25)
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{1}\asymp
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{1}\asymp
2^{-nr_{1}}n^{(\nu-1)(\frac{1}{p^{*}}-\frac{1}{\theta})_{+}}.$

In the case $1<p\leq 2$, Theorem G was proved in [43] for $\theta=\infty$, and
in [12, 28] for $1\leq\theta<\infty$.

For $2<p\leq\infty$ and $\theta=\infty$, the upper bounds in Theorem G follow
from the upper bounds for
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(H^{\boldsymbol{r}}_{2})_{2}$,
see [5]. The lower bounds are nontrivial. They follow from the corresponding
lower bounds for the Kolmogorov widths
$d_{M}(H^{\boldsymbol{r}}_{\infty},L_{1})$, which, as it was observed in [2],
follow from the lower bounds for the entropy numbers
$\varepsilon_{M}(H^{\boldsymbol{r}}_{\infty},L_{1})$ from [44].

If $2<p\leq\infty$ and $1\leq\theta<\infty$, the upper estimate for the
quantity
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{1}$
follows from Theorem B as $p=2$. It is due to the relation
$\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{1}\ll\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{2,\theta})_{2}$,
$2<p\leq\infty$.

The lower estimates for the quantity
$E_{Q^{{\boldsymbol{\gamma}^{\prime}}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{1}$,
$2<p\leq\infty$, in the case $1\leq\theta\leq 2$ were obtained in [28], and
for $2<\theta<\infty$ they follow from the estimate for the Kolmogorov width
$d_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{1})$, see [31].

######  Remark 3.11.

Comparing the estimates (3.23) with (3.24) – (3.25), we conclude that the
corresponding approximation characteristics for the classes
$B^{\boldsymbol{r}}_{p,\theta}$, $r_{1}>0$, $1\leq\theta\leq\infty$ in the
spaces $B_{q,1}$ and $L_{q}$ for $1\leq q<p\leq\infty$ and $d\geq 2$ coincide
in order only either in the case $\theta=1$ or $\nu=1$.

######  Remark 3.12.

It is worth noting, that in the univariate case these approximation
characteristics in the spaces $B_{q,1}$ and $L_{q}$ coincide in order for all
values of the parameters $p$, $q$ and $\theta$, see [35].

##  4\. Widths and entropy numbers

In this part of the paper, the main attention is paid to getting the exact
order estimates for the Kolmogorov widths and entropy numbers for the classes
$B^{\boldsymbol{r}}_{p,\theta}$ in the space $B_{q,1}$. In addition, using the
obtained estimates for the Kolmogorov widths in combination with the results
from Section 3, we write down the exact order estimates for the linear widths
of the investigated classes of functions in the same space.

As a result of research, we found out that the orders of the Kolmogorov and
linear widths in the considered cases are realized by the subspace of
trigonometric polynomials with spectrum form the step hyperbolic crosses.

Let us define the corresponding approximation characteristics.

Let $\mathscr{Y}$ be a normed space with the norm $\|\cdot\|_{\mathscr{Y}}$,
$\mathscr{L}_{M}(\mathscr{Y})$ be a set of subspaces of dimension at most $M$
in the space $\mathscr{Y}$ , and $W$ be a centrally-symmetric set in
$\mathscr{Y}$.

The quantity

$d_{M}(W,\mathscr{Y}):=\inf\limits_{L_{M}\in\mathscr{L}_{M}(\mathscr{Y})}\sup\limits_{w\in
W}\inf\limits_{u\in L_{M}}\|w-u\|_{\mathscr{Y}}$

is called the Kolmogorov $M$-width of the set $W$ in the space $\mathscr{Y}$.

The width $d_{M}(W,\mathscr{Y})$ was introduced in 1936 by A.N. Kolmogorov
[18].

Let $\mathscr{Y}$ and $\mathscr{Z}$ be normed spaces and
$L(\mathscr{Y},\mathscr{Z})$ be a set of linear continuous mappings of
$\mathscr{Y}$ into $\mathscr{Z}$.

The quantity

$\lambda_{M}(W,\mathscr{Y}):=\mathop{\inf_{L_{M}\in\mathscr{L}_{M}(\mathscr{Y})}}\limits_{\Lambda\in
L(\mathscr{Y},L_{M})}\sup\limits_{w\in W}\|w-\Lambda w\|_{\mathscr{Y}},$

where $L(\mathscr{Y},L_{M})$ denotes linear continuous operators that map from
$\mathscr{Y}$ into $L_{M}$, is called the linear $M$-width. It was introduced
in 1960 by V.M. Tikhomirov [50].

From the definitions of widths, it follows that

(4.1) $d_{M}(F,\mathscr{Y})\leq\lambda_{M}(F,\mathscr{Y}).$

The history of investigation of widths for different functional classes of
periodic multivariate functions is given in the monographs [8, 30, 42, 45, 48,
51]

Let further $\mathscr{X}$ be a Banach space and let $B_{\mathscr{X}}(y,R)$ be
a ball in $\mathscr{X}$ of radius $R$ centered at a point $y$, i.e.,

$B_{\mathscr{X}}(y,R):=\\{x\in\mathscr{X}\colon\,\,\|{x}-y\|\leq R\\}.$

For a compact set $A$ and $\varepsilon>0$, we introduce the number

(4.2) $N_{\varepsilon}(A,\mathscr{X}):=\min\Big{\\{}n\colon\quad\exists
y^{1},\ldots,y^{n}\in\mathscr{X}\colon\quad
A\subseteq\bigcup\limits^{n}_{j=1}B_{\mathscr{X}}(y^{j},\varepsilon)\Big{\\}}.$

Then the quantity (see. e.g., [19])

$H_{\varepsilon}(A,\mathscr{X}):=\log N_{\varepsilon}(A,\mathscr{X})$

is called the $\varepsilon$-entropy of the set $A$ with respect to the Banach
space $\mathscr{X}$ (here and in what follows, $\log:=\log_{2}$).

The $\varepsilon$-entropy of the set $A$ is closely connected to its entropy
numbers $\varepsilon_{k}(A,\mathscr{X})$ (see. e.g., [14]):

$\varepsilon_{k}(A,\mathscr{X}):=\inf\Big{\\{}\varepsilon\colon\quad\exists
y^{1},\ldots,y^{2^{k}}\in\mathscr{X}\colon\quad
A\subseteq\bigcup\limits^{2^{k}}_{j=1}B_{\mathscr{X}}(y^{j},\varepsilon)\Big{\\}}.$

Note that directly from the definitions of the quantities
$H_{\varepsilon}(A,\mathscr{X})$ and $\varepsilon_{k}(A,\mathscr{X})$, we
conclude that if $H_{\varepsilon}(A,\mathscr{X})\leq k$ then
$\varepsilon_{k}(A,\mathscr{X})\leq\varepsilon$ and, vice versa, the estimate
$\varepsilon_{k}(A,\mathscr{X})\leq\varepsilon$ yields the estimate
$H_{\varepsilon}(A,\mathscr{X})\leq k$.

In other words, if $k<H_{\varepsilon}(A,\mathscr{X})\leq k+1$ then
$\varepsilon_{k+1}(A,\mathscr{X})\leq\varepsilon\leq\varepsilon_{k}(A,\mathscr{X})$.
These relations enable one to establish estimates for the
$\varepsilon$-entropy $H_{\varepsilon}(A,\mathscr{X})$ from the estimates for
the corresponding entropy numbers $\varepsilon_{k}(A,\mathscr{X})$.

The above defined characteristics on the classes of periodic multivariate
functions $W^{\boldsymbol{r}}_{p,\alpha}$, $H^{\boldsymbol{r}}_{p}$ and
$B^{\boldsymbol{r}}_{p,\theta}$ in the space $L_{q}$ were investigated in a
number of papers. The detailed bibliography can be found in the monographs [8,
48, 51].

In addition, we highlight the recent papers [21, 32, 49], where the questions
concerning estimates of the Kolmogorov width and entropy numbers of the
mentioned classes of functions in the space $L_{q}$ were investigated, and
also [16, 17, 36, 38], where similar questions were considered on the classes
of quasi-continuous functions $QC$. The norm in this space is close in some
sense to the $L_{\infty}$-norm, and it is connected in a certain way with the
$B_{\infty,1}$-norm (see, e.g., [17] for the details).

The next statement will play an important role in obtaining the estimates for
the entropy numbers and Kolmogorov width of the classes
$B^{\boldsymbol{r}}_{p,\theta}$ in the space $B_{q,1}$. It is a corollary from
the Carl’s inequality (see, e.g., [6]).

Lemma B. [47]. Let $A$ be a compact set in a separable Banach space
$\mathscr{X}$. Assume that for a pair of numbers $(a,b)$, where either $a>0$,
$b\in\mathbb{R}$ or $a=0$, $b<0$, the relations

$d_{m}(A,\mathscr{X})\ll m^{-a}(\log
m)^{b},\qquad\varepsilon_{m}(A,\mathscr{X})\gg m^{-a}(\log m)^{b}$

are true. Then, it holds

$\varepsilon_{m}(A,\mathscr{X})\asymp d_{m}(A,\mathscr{X})\asymp m^{-a}(\log
m)^{b}.$

Before we formulate and prove the results, let us introduce some further
notation, taken from [44].

For all ${\boldsymbol{s}}\in\mathbb{N}^{d}$, let us define

$\overline{\rho}({\boldsymbol{s}}):=\\{\boldsymbol{k}\in\mathbb{N}^{d}\colon\
2^{s_{j}-1}\leq k_{j}<2^{s_{j}},\ j=1,\dots,d\\},$

and consider the set of real trigonometric polynomials

$T(\overline{\rho}({\boldsymbol{s}})):=\Big{\\{}t\colon\
t(\boldsymbol{x})=\sum\limits_{\boldsymbol{k}\in\overline{\rho}({\boldsymbol{s}})}\widehat{t}(\boldsymbol{k})e^{i(\boldsymbol{k},\boldsymbol{x})},\
\boldsymbol{x}\in\mathbb{R}^{d}\Big{\\}},$

where $\widehat{t}(\boldsymbol{k})$ are the Fourier coefficients of $t$.

Every $t\in T(\overline{\rho}({\boldsymbol{s}}))$, $s_{j}\geq 2$,
$j=1,\dots,d$, can be represented in the form

$t(\boldsymbol{x})=e^{i(\boldsymbol{k}^{\boldsymbol{s}},\boldsymbol{x})}t^{1}(\boldsymbol{x}),$

where $\boldsymbol{k}^{\boldsymbol{s}}=(k^{s_{1}}_{1},\ldots,k^{s_{d}}_{d})$,
$k^{s_{j}}_{j}=2^{s_{j}-1}+2^{s_{j}-2}$, $j=1,\dots,d$, and
$t^{1}(\boldsymbol{x})$ is a polynomial of degree $2^{s_{j}-2}$ with respect
to the variable $x_{j}$, $j=1,\dots,d$.

For every ${\boldsymbol{m}}\in\mathbb{Z}_{+}^{d}$, i.e., with nonnegative
integer coordinates, we put

$RT({\boldsymbol{m}}):=\Bigg{\\{}t\colon\quad
t(\boldsymbol{x})=\mathop{\sum_{|k_{j}|\leq
m_{j}}}\limits_{j=1,\dots,d}\widehat{t}(\boldsymbol{k})e^{i(\boldsymbol{k},\boldsymbol{x})},\
\boldsymbol{x}\in\mathbb{R}^{d}\Bigg{\\}}.$

Further, let

$T^{\prime}(\overline{\rho}({\boldsymbol{s}})):=\Big{\\{}t\colon\
t(\boldsymbol{x})=e^{i(\boldsymbol{k}^{\boldsymbol{s}},\boldsymbol{x})}t^{1}(\boldsymbol{x}),\,\,t^{1}\in
RT(2^{{\boldsymbol{s}}-2})\Big{\\}}.$

For even $n\in\mathbb{N}$, we define the sets

$\displaystyle\Omega_{n}:=\\{{\boldsymbol{s}}\in\mathbb{N}^{d}\colon\,(\boldsymbol{s},\boldsymbol{1})=n,\,s_{j}\mbox{
are even numbers},\,j=1,\dots,d\\},\
Q^{\prime}_{n}:=\bigcup\limits_{{\boldsymbol{s}}\in\Omega_{n}}\overline{\rho}({\boldsymbol{s}}),$
$\displaystyle
T^{\prime}(Q^{\prime}_{n}):=\bigcup\limits_{{\boldsymbol{s}}\in\Omega_{n}}T^{\prime}(\overline{\rho}({\boldsymbol{s}}))=\\{t\colon\quad
t(\boldsymbol{x})=\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}e^{i(\boldsymbol{k}^{\boldsymbol{s}},\boldsymbol{x})}t^{1}_{\boldsymbol{s}}(\boldsymbol{x}),\quad
t^{1}_{\boldsymbol{s}}\in RT(2^{{\boldsymbol{s}}-2})\\},$ $\displaystyle
T^{\prime}(Q^{\prime}_{n})_{\infty}:=\\{t\in
T^{\prime}(Q^{\prime}_{n})\colon\quad\|t^{1}_{\boldsymbol{s}}\|_{\infty}\leq
1\\}.$

In what follows, we get the lower estimate for the entropy numbers of the
classes $B^{\boldsymbol{r}}_{\infty,\theta}$, $1\leq\theta\leq\infty$, in the
space $B_{1,1}$. It is an expansion of the corresponding result by V.N.
Temlyakov [44] for the classes $H^{\boldsymbol{r}}_{\infty}$.

######  Theorem 4.1.

Let $r_{1}>0$ and $1\leq\theta\leq\infty$. Then it holds

$\varepsilon_{M}(B^{\boldsymbol{r}}_{\infty,\theta},B_{1,1})\gg
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+1-\frac{1}{\theta}}.$

###### Proof.

In the case $\theta=\infty$, as it was noted above, the estimate was obtained
in [44]. Therefore, further we assume that $1\leq\theta<\infty$. Besides, it
is sufficient to consider the case $\nu=d$.

We will use some relations from the proof of Theorem 2.2 in [44], in
particular, the estimate

(4.3)
$\varepsilon_{M}(T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}},L_{2})\gg
M^{-r_{1}}(\log^{d-1}M)^{r_{1}+\frac{1}{2}},$

where $M=|Q^{{}^{\prime}}_{n}|\asymp 2^{n}n^{d-1}$.

Since the embedding

$T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}}\subset
C_{7}H^{\boldsymbol{r}}_{\infty},\quad C_{7}>0,$

is true, for $f\in T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}}$ it holds
$\|A_{\boldsymbol{s}}(f)\|_{\infty}\ll 2^{-({\boldsymbol{s}},r)}$,
$s\in\Omega_{n}$ (see (2.8)).

Therefore, for $f\in T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}}$ from
(2.7) and (3.18) we get

$\displaystyle\|f\|_{B^{\boldsymbol{r}}_{\infty,\theta}}$
$\displaystyle\asymp\Bigg{(}\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|A_{\boldsymbol{s}}(f)\|^{\theta}_{\infty}\Bigg{)}^{\frac{1}{\theta}}\ll\Bigg{(}\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\|A_{\boldsymbol{s}}\|^{\theta}_{1}\,\,\|f\|^{\theta}_{\infty}\Bigg{)}^{\frac{1}{\theta}}$
$\displaystyle\ll\Bigg{(}\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}2^{({\boldsymbol{s}},{\boldsymbol{r}})\theta}\,\,\|f\|^{\theta}_{\infty}\Bigg{)}^{\frac{1}{\theta}}\ll\Bigg{(}\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}1\Bigg{)}^{\frac{1}{\theta}}\asymp
n^{\frac{d-1}{\theta}},$

and thus

(4.4)
$T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}}n^{-\frac{d-1}{\theta}}\subset
C_{8}B^{\boldsymbol{r}}_{\infty,\theta},\quad 1\leq\theta<\infty,\quad
C_{8}>0.$

Hence, according to (4.4) and (4.3), in view of $M\asymp 2^{n}n^{d-1}$, i.e.,
$n\asymp\log M$, we can write

$\displaystyle\varepsilon_{M}(T^{\prime}(Q^{\prime}_{n})_{\infty}\,\,2^{-nr_{1}}\,\,n^{-\frac{d-1}{\theta}},L_{2})$
$\displaystyle\gg
M^{-r_{1}}(\log^{d-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}$ (4.5)
$\displaystyle\asymp 2^{-nr_{1}}n^{(d-1)(\frac{1}{2}-\frac{1}{\theta})}.$

Let $A$ be a compact set and $M_{\varepsilon}(A,\mathscr{X})$ be the maximum
number of points $x^{j}\in A$ such that
$\|x^{i}-x^{j}\|_{\mathscr{X}}>\varepsilon$, $i\neq j$;
$N_{\varepsilon}(A,\mathscr{X})$ be a number defined by the formula (4.2).

It is well known that

(4.6) $N_{\varepsilon}(A,\mathscr{X})\leq M_{\varepsilon}(A,\mathscr{X})\leq
N_{\varepsilon/2}(A,\mathscr{X}).$

The relations (4.5) and (4.6) yield that in
$T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}}n^{-\frac{d-1}{\theta}}$
there exist $2^{M}$ functions $\\{f_{j}\\}^{2^{M}}_{j=1}$ such that for $i\neq
j$ the following estimate is true:

(4.7) $\|f_{i}-f_{j}\|_{2}\gg
2^{-nr_{1}}n^{(d-1)(\frac{1}{2}-\frac{1}{\theta})}.$

Further we show that

(4.8) $\|f_{i}-f_{j}\|_{B_{1,1}}\gg 2^{-nr_{1}}n^{(d-1)(1-\frac{1}{\theta})}.$

Indeed, taking into account (4.7) and the definition (2.9) of the norm in the
space $B_{1,1}$, for $f\in T^{\prime}(Q^{\prime}_{n})_{\infty}$ we obtain

$\displaystyle n^{d-1}\ll\|f\|^{2}_{2}$
$\displaystyle=\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}\,\|\sum\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}A_{{\boldsymbol{s}}^{\prime}}(f)\|^{2}_{2}\leq\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}\sum\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}\|A_{{\boldsymbol{s}}^{\prime}}(f)\|^{2}_{2}$
$\displaystyle\leq\sum\limits_{{\boldsymbol{s}}\in\Omega_{n}}\sum\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}\|A_{{\boldsymbol{s}}^{\prime}}(f)\|_{1}\,\|A_{{\boldsymbol{s}}^{\prime}}(f)\|_{\infty}$
(4.9)
$\displaystyle\leq\mathop{\max_{{\boldsymbol{s}}\in\Omega_{n}}}\limits_{\|{\boldsymbol{s}}-{\boldsymbol{s}}^{\prime}\|_{\infty}\leq
1}\|A_{{\boldsymbol{s}}^{\prime}}(f)\|_{\infty}\sum\limits_{{\boldsymbol{s}}\in\mathbb{N}^{d}}\|A_{{\boldsymbol{s}}}(f)\|_{1}\ll\|f\|_{B_{1,1}}.$

Combining (4.7) with (4.9), we get (4.8).

Hence, from (4.8), in view of $M\asymp 2^{n}n^{d-1}$, we obtain the estimate

$\displaystyle\varepsilon_{M}(B^{\boldsymbol{r}}_{\infty,\theta},B_{1,1})$
$\displaystyle\gg\varepsilon_{M}(T^{\prime}(Q^{\prime}_{n})_{\infty}\,2^{-nr_{1}}n^{-\frac{d-1}{\theta}},B_{1,1})\gg
2^{-nr_{1}}n^{(d-1)(1-\frac{1}{\theta})}$ $\displaystyle\asymp
M^{-r_{1}}(\log^{d-1}M)^{r_{1}+1-\frac{1}{\theta}}.$

Theorem 4.1 is proved. ∎

In what follows, we prove the estimates for the Kolmogorov widths and entropy
numbers of the classes $B^{\boldsymbol{r}}_{p,\theta}$ in the space $B_{q,1}$.

######  Theorem 4.2.

Let $d\geq 2$, $r_{1}>0$, $1\leq q\leq p\leq\infty$, $1\leq\theta\leq\infty$.
Then it holds

(4.10) $\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})\asymp
d_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+1-\frac{1}{\theta}}.$

###### Proof.

In order to use Lemma B, let us note the following.

From the one hand, the upper estimate for the width
$d_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})$ follows from Theorems 3.2, 3.5
and 3.10 under the condition $M\asymp 2^{n}n^{\nu-1}$, i.e.,

$\displaystyle d_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})$ $\displaystyle\ll
E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}$ (4.11) $\displaystyle\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+1-\frac{1}{\theta}}.$

From the other hand, the lower estimate for the entropy numbers
$\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})$ is a corollary from
Theorem 4.1, i.e.,

$\displaystyle\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})$
$\displaystyle\geq\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{1,1})\geq\varepsilon_{M}(B^{\boldsymbol{r}}_{\infty,\theta},B_{1,1})$
(4.12) $\displaystyle\gg
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+1-\frac{1}{\theta}}.$

Therefore, taking Lemma B into account with respect to the relations (4.11)
and (4.12), we obtain (4.10).

Theorem 4.2 is proved. ∎

######  Remark 4.3.

The estimate (4.10) for the Kolmogorov width yields the lower bounds for
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{\infty,\theta})_{B_{\infty,1}}$
in Theorem 3.5 and for
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}$,
$1\leq q<p\leq\infty$, in Theorem 3.10. Indeed, for $M\asymp 2^{n}n^{\nu-1}$
it holds

$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{q,1}}\gg
d_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})\asymp
2^{-nr_{1}}n^{(\nu-1)(1-\frac{1}{\theta})}.$

Further we compare the result of Theorem 4.2 with the estimates of the
corresponding quantities in the space $L_{q}$. We will consider several cases.

First, let $p=q$.

######  Remark 4.4.

In the case $p=1$, the question about the orders of the quantities
$d_{M}(B^{\boldsymbol{r}}_{1,\theta},L_{1})$ and
$\varepsilon_{M}(B^{\boldsymbol{r}}_{1,\theta},L_{1})$ for $d\geq 2$ remains
open. In particular, for the width $d_{M}(H^{\boldsymbol{r}}_{1},L_{1})$ (see,
e.g., [8, Figure 4.3]).

Case $p=\infty$. Here the following statements are known for the case $d=2$.

Theorem H. [32] Let $d=2$, ${\boldsymbol{r}}=(r_{1},r_{1})$, $r_{1}>0$,
$1\leq\theta<\infty$. Then it holds

(4.13) $\varepsilon_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})\asymp
d_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})\asymp M^{-r_{1}}(\log
M)^{r_{1}+1-\frac{1}{\theta}}.$

Theorem I. Let $d=2$, ${\boldsymbol{r}}=(r_{1},r_{1})$, $r_{1}>0$. Then it
holds

(4.14) $\varepsilon_{M}(H^{\boldsymbol{r}}_{\infty},L_{\infty})\asymp
d_{M}(H^{\boldsymbol{r}}_{\infty},L_{\infty})\asymp M^{-r_{1}}(\log
M)^{r_{1}+1}.$

The upper estimate for $d_{M}(H^{\boldsymbol{r}}_{\infty},L_{\infty})$ is
realized by approximating functions from the classes
$H^{\boldsymbol{r}}_{\infty}$ by trigonometric polynomials with harmonics from
the hyperbolic cross [42, p. 55]. For the corresponding lower estimate for the
entropy numbers $\varepsilon_{M}(H^{\boldsymbol{r}}_{\infty},L_{\infty})$, see
[46]. Hence, applying Lemma B, we get (4.14).

######  Remark 4.5.

Comparing the estimates (4.13) and (4.14) with the result of Theorem 4.2 for
$\nu=2$ and $p=\infty$, we see that the orders of the respective
characteristics of the classes $B^{\boldsymbol{r}}_{\infty,\theta}$ in the
spaces $B_{\infty,1}$ and $L_{\infty}$ coincide.
The question about the orders of the quantities
$d_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})$ and
$\varepsilon_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})$ in the case
$d\geq 3$ and $1<\theta\leq\infty$ still remains open.

To complement the already mentitoned results, we further formulate the known
statement for the quantity $d_{M}(B^{\boldsymbol{r}}_{\infty,1},L_{\infty})$
in the case $d\geq 2$.

Theorem J. [27] Let $d\geq 2$, $r_{1}>0$. Then it holds

(4.15) $d_{M}(B^{\boldsymbol{r}}_{\infty,1},L_{\infty})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}}.$

######  Remark 4.6.

Comparing (4.15) with (4.10) for $\theta=1$ and $p=\infty$, we conclude that

$d_{M}(B^{\boldsymbol{r}}_{\infty,1},B_{\infty,1})\asymp
d_{M}(B^{\boldsymbol{r}}_{\infty,1},L_{\infty})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}}.$

Case $2\leq p<\infty$. In what follows, we prove the statement which is an
analog to Theorem 4.2 as $p=q$ for the space $L_{p}$ under certain
restrictions on the parameters $p$ and $\theta$.

######  Theorem 4.7.

Let $d\geq 2$, $r_{1}>0$, $2\leq p<\infty$, $2\leq\theta\leq\infty$. Then it
holds

(4.16) $\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})\asymp
d_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}.$

###### Proof.

Let first $2\leq\theta<\infty$. Then, from the one hand, Theorem B as $M\asymp
2^{n}n^{\nu-1}$ yields

$\displaystyle d_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})$
$\displaystyle\ll\mathscr{E}_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{p}\asymp
2^{-nr_{1}}n^{(\nu-1)(\frac{1}{2}-\frac{1}{\theta})}$ (4.17)
$\displaystyle\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}.$

From the other hand, according to [31, Thm. 3], it holds

(4.18)
$\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})\geq\varepsilon_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{1})\gg
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}.$

Using Lemma B with respect to the estimates (4.17) and (4.18), we get

$\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})\asymp
d_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}},\quad
2\leq\theta<\infty.$

Let $\theta=\infty$. Then we obtain (4.16) by using Lemma B with respect to
the estimates

$\varepsilon_{M}(H^{\boldsymbol{r}}_{\infty},L_{1})\gg
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}}$

and

$d_{M}(H^{\boldsymbol{r}}_{p},L_{p})\ll
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}}$

(see, e.g., [43] and [42, Ch. 3, $\S$ 2], respectively).

Theorem 4.7 is proved. ∎

Let further $p\neq q$. The following statement is known.

Theorem K [31]. Let $d\geq 2$, $r_{1}>0$, $2\leq p\leq\infty$, $1\leq q<p$,
$2\leq\theta\leq\infty$. Then it holds

(4.19) $\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{q})\asymp
d_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{q})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}.$

In the case $\theta=\infty$, the estimates for the quantity
$d_{M}(H^{\boldsymbol{r}}_{p},L_{q})$ with further comments can be found in
[8, Thms. 4.3.10 – 4.3.13]. The lower estimate for the entropy numbers
$\varepsilon_{M}(H^{\boldsymbol{r}}_{\infty},L_{1})$ was obtained in [44].

######  Remark 4.8.

Comparing (4.10) with (4.16) and (4.19), we conclude the following. Under the
conditions of Theorem 4.7 and Theorem K on the parameters $p$, $q$ and
$\theta$, the corresponding approximation characteristics of the classes
$B^{\boldsymbol{r}}_{p,\theta}$ in the spaces $L_{q}$ and $B_{q,1}$ differ in
order.
For the remaining values of the parameters $p$, $q$ and $\theta$, that fulfil
the conditions of Theorem 4.2, the orders of the quantities
$d_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{q})$ and
$\varepsilon_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{q})$, to the best of our
knowledge, remain open.

Finally, we formulate two corollaries that concern the linear widths of the
classes $B^{\boldsymbol{r}}_{p,\theta}$ in the space $B_{q,1}$, and give
further comments, comparing the obtained results with the corresponding
statements in the space $L_{q}$.

######  Corollary 4.9.

Let $d\geq 2$, $r_{1}>0$, $1\leq q\leq p\leq\infty$, $1\leq\theta\leq\infty$.
Then it holds

(4.20) $\lambda_{M}(B^{\boldsymbol{r}}_{p,\theta},B_{q,1})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+1-\frac{1}{\theta}}.$

###### Proof.

The upper estimate in (4.20) follows from Theorems 3.2, 3.5 and 3.10 as
$M\asymp 2^{n}n^{\nu-1}$. Here we note, that for
$(p,q)\in\\{(1,1),(\infty,\infty)\\}$ the upper estimate for the quantity of
the best approximation
$E_{Q^{{\boldsymbol{\gamma}}^{\prime}}_{n}}(B^{\boldsymbol{r}}_{p,\theta})_{B_{p,1}}$
from Theorem 3.5 is realized by a linear method, based on the polynomials
(3.16).

The corresponding lower estimate, according to the relation (4.1), is a
corollary from Theorem 4.2. ∎

In what follows, we comment on the corresponding results in the spaces $L_{p}$
and $L_{q}$. Here we consider several cases.

######  Remark 4.10.

For $p=q=1$, the orders of the quantity
$\lambda_{M}(B^{\boldsymbol{r}}_{1,\theta},L_{1})$ in the case $d\geq 2$ are
not known (see, e.g., [8, Figure 4.5]).

Case $p=q=\infty$. Let us formulate a corollary of the known results.

######  Corollary 4.11.

Let $d=2$, ${\boldsymbol{r}}=(r_{1},r_{1})$, $r_{1}>0$,
$1\leq\theta\leq\infty$. Then it holds

(4.21) $\lambda_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})\asymp
M^{-r_{1}}(\log M)^{r_{1}+1-\frac{1}{\theta}}.$

###### Proof.

The upper estimate for
$\lambda_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})$ is realized by
approximating functions from the classes $B^{\boldsymbol{r}}_{\infty,\theta}$
by trigonometric polynomials with harmonics from the hyperbolic crosses (see
[42, p. 55] for the case $\theta=\infty$ and [29] for $1\leq\theta<\infty$).

The lower estimate in (4.21) follows from (4.13) and (4.14). ∎

######  Remark 4.12.

Comparing (4.21) with (4.20) for $\nu=2$ and $p=\infty$, we conclude that for
${\boldsymbol{r}}=(r_{1},r_{1})$, $r_{1}>0$, $1\leq\theta\leq\infty$, it holds

$\lambda_{M}(B^{\boldsymbol{r}}_{\infty,\theta},B_{\infty,1})\asymp\lambda_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})\asymp
M^{-r_{1}}(\log M)^{r_{1}+1-\frac{1}{\theta}}.$

In the case $d\geq 3$ and $1<\theta\leq\infty$, the question about the order
of the quantity $\lambda_{M}(B^{\boldsymbol{r}}_{\infty,\theta},L_{\infty})$
remains open.

To complement (4.21), we formulate a corollary from Theorem J (in view of the
fact that the upper estimate for the respective Kolmogorov width is realized
by a linear method).

######  Corollary 4.13.

Let $d\geq 2$. Then for $r_{1}>0$ it holds

(4.22) $\lambda_{M}(B^{\boldsymbol{r}}_{\infty,1},L_{\infty})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}}.$

######  Remark 4.14.

Comparing (4.22) with (4.20) for $p=\infty$ and $\theta=1$, we get that in the
case $d\geq 2$ it holds

$\lambda_{M}(B^{\boldsymbol{r}}_{\infty,1},B_{\infty,1})\asymp\lambda_{M}(B^{\boldsymbol{r}}_{\infty,1},L_{\infty})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}}.$

Case $2\leq p=q<\infty$. From Theorems B and 4.7, we obtain the following.

######  Corollary 4.15.

Let $d\geq 2$, $r_{1}>0$, $2\leq p<\infty$, $2\leq\theta\leq\infty$. Then it
holds

(4.23) $\lambda_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{p})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}.$

Case $p\neq q$. From Theorems F and K, we obtain the following.

######  Corollary 4.16.

Let $d\geq 2$, $r_{1}>0$, $2\leq p\leq\infty$, $1\leq q<p$,
$2\leq\theta\leq\infty$. Then it holds

(4.24) $\lambda_{M}(B^{\boldsymbol{r}}_{p,\theta},L_{q})\asymp
M^{-r_{1}}(\log^{\nu-1}M)^{r_{1}+\frac{1}{2}-\frac{1}{\theta}}.$

######  Remark 4.17.

Comparing (4.23) and (4.24) with (4.20), we conclude that under the conditions
of Corollaries 4.15 and 4.16 on the parameters $p$, $q$ and $\theta$, the
corresponding approximation characteristics in the spaces $B_{q,1}$ and
$L_{q}$ differ in order.

Acknowledgement. KP would like to acknowledge support by the Philipp Schwartz
Fellowship of the Alexander von Humboldt Foundation.

## References

  * [1] T.I. Amanov, Representation and imbedding theorems for function spaces $S^{(r)}_{p,\theta}B(\mathbb{R}_{n})$ and $S^{(r)}_{p,\theta}\ast B$, $(0\leq x_{j}\leq 2\pi$; $j=1,\ldots,n)$, Tr. Mat. Inst. Akad. Nauk SSSR 77 (1965) 5–34 (in Russian).
  * [2] E.S. Belinskii, Asymptotic characteristics of classes of functions with conditions on the mixed derivative (mixed difference). In: Studies in the theory of functions of several real variables, Yaroslav State Univ. (Y. Brudnyi, Ed.), Yaroslavl (1990) 22–37 (in Russian).
  * [3] E.S. Belinsky, Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative, J. Approx. Theory 93 (1998) 114–127.
  * [4] O.V. Besov, Investigation of one family of functional spaces in connection with the embedding and continuation theorems, Tr. Mat. Inst. Acad. Nauk SSSR 60 (1961) 42–81 (in Russian).
  * [5] Ya.S. Bugrov, Approximation of a class of functions with dominant mixed derivative, Sb. Math. (N.S.) 106 (3) (1964) 410–418 (in Russian).
  * [6] B. Carl, Entropy numbers, $s$-numbers, and eigen-value problems, J. Funct. Anal. 41 (1981) 290–306.
  * [7] D. Dũng, Approximation of classes of functions on the torus defined by a mixed modulus of continuity. In: Constructive theory of functions (Proc. Internat. Conf., Varna, 1984). Bulgarian Academy of Science, Sofia, 1984, 43–48 (in Russian).
  * [8] D. Dũng, V. Temlyakov, T. Ullrich, Hyperbolic cross approximation, Adv. Courses Math. Birkhäuser, CRM Barselona, 2018.
  * [9] O.V. Fedunyk-Yaremchuk, M.V. Hembars’kyi, S.B. Hembars’ka, Approximative characteristics of the Nikol’skii-Besov-type classes of periodic functions in the space $B_{\infty,1}$, Carpathian Math. Publ. 12 (2) (2020) 376–391.
  * [10] O.V. Fedunyk-Yaremchuk, S.B. Hembars’ka, Best orthogonal trigonometric approximations of the Nikol’skii-Besov-type classes of periodic functions of one and several variables, Carpathian Math. Publ. 14 (1) (2022) 171–184.
  * [11] E.M. Galeev, Kolmogorov widths of certain classes of periodic functions of several variables, In: Constructive theory of functions (Proc. Internat. Conf., Varna, 1984). Bulgarian Academy of Science, Sofia, 1984, 27–32 (in Russian).
  * [12] E.M. Galeev, Approximation of classes of periodic functions of several variables by nuclear operator, Math. Notes 47 (3) (1990) 248–254.
  * [13] S.B. Hembars’ka, P.V. Zaderei, Best orthogonal trigonometric approximations of the Nikol’skii–Besov-type classes of periodic functions in the space $B_{\infty,1}$, Ukr. Math. J. 74 (6) (2022) 883–895.
  * [14] K. Höllig, Diameters of classes of smooth functions. In: Quantitative Approximation. Academic Press, New York, 163–176.
  * [15] B.S. Kashin, V.N. Temlyakov, On best $m$-term approximations and the entropy of sets in the space $L^{1}$, Math. Notes 56 (5) (1994) 1137–1157.
  * [16] B.S. Kashin, V.N. Temlyakov, On a certain norm and related applications, Math. Notes 64 (4) (1999) 551–554.
  * [17] B.S. Kashin, V.N. Temlyakov, On a norm and approximation characteristics of classes of functions of several variables. In: Metric theory of functions and related problems in analysis. Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra (AFTs), Moscow, 1999, 69–99 (in Russian).
  * [18] A. Kolmogoroff, Über die beste Annäherung von Fuktionen einer gegebenen Funktionenklasse, Ann. Math. 37 (1) (1936) 107–110.
  * [19] A.N. Kolmogorov, V.M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces, Uspekhi Mat. Nauk, 14 (2) (1959) 3–86 (in Russian).
  * [20] P.I. Lizorkin, S.M. Nikol’skii, Spaces of functions of mixed smoothness from the decomposition point of view, Tr. Mat. Inst. Acad. Nauk SSSR 187 (3) (1989) 143–161 (in Russian).
  * [21] S. Mayer, T. Ullrich, Entropy numbers of finite dimensional mixed-norm balls and function space embeddings with small mixed smoothness, Constr. Approx. 53 (2021) 249–279.
  * [22] S.M. Nikol’skii, Inequalities for entire functions of finite power and their application to the theory of differentiable functions of many variables, Tr. Mat. Inst. Akad. Nauk SSSR 38 (1951) 244–278 (in Russian).
  * [23] S.M. Nikol’skii, Functions with dominant mixed derivative, satisfying a multiple Holder condition, Sibirsk. Mat. Zh. 4 (6) (1963) 1342–1364 (in Russian).
  * [24] N.S. Nikol’skaya, The approximation in the $L_{p}$ metric of differentiable functions of several variables by Fourier sums, Sib. Math. J. 15 (2) (1974) 282–295.
  * [25] A.S. Romanyuk, Approximation of the Besov classes of periodic functions of several variables in a space $L_{q}$, Ukr. Math. J. 43 (10) (1991) 1297–1306.
  * [26] A.S. Romanyuk, Approximability of the classes $B^{\boldsymbol{r}}_{p,\theta}$ of periodic functions of several variables by linear methods and best approximations, Sb. Math. 195 (2) (2004) 237–261.
  * [27] A.S. Romanyuk, Kolmogorov widths of Besov classes $B^{\boldsymbol{r}}_{p,\theta}$ in the metric of the space $L_{\infty}$, Ukr. Math. Bull., 2 (2) (2005) 205–222.
  * [28] A.S. Romanyuk, Best approximations and widths of classes of periodic functions of several variables, Sb. Math. 199 (2) (2008) 253–275.
  * [29] A.S. Romanyuk, Diameters and best approximation of the classes $B^{\boldsymbol{r}}_{p,\theta}$ of periodic functions of several variables, Anal. Math. 37 (2011) 181–213.
  * [30] A.S. Romanyuk, Approximative charecteristics of the classes of periodic functions of many variables. In: Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Vol. 93, 2012 (in Russian).
  * [31] A.S. Romanyuk, Estimation of the entropy numbers and Kolmogorov widths for the Nikol’skii–Besov classes of periodic functions of many variables, Ukr. Math. J. 67 (11) (2016) 1739–1757.
  * [32] A.S. Romanyuk, Entropy numbers and widths for the Nikol’skii-Besov classes of functions of many variables in the space $L_{\infty}$, Anal. Math. 45 (1) (2019) 133–151.
  * [33] A.S. Romanyuk, V.S. Romanyuk, Estimation of some approximating characteristics of the classes of periodic functions of one and many variables, Ukr. Math. J. 71 (8) (2020) 1257–1272.
  * [34] A.S. Romanyuk, V.S. Romanyuk, Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii-Besov spaces, J. Math. Sci. 252 (4) (2021) 508–525.
  * [35] A.S. Romanyuk, V.S. Romanyuk, K.V. Pozharska, S.B. Hembars’ka, Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions, Carpathian Math. Publ. 15 (1) (2023) 78–94.
  * [36] A.S. Romanyuk, S.Ya. Yanchenko, Kolmogorov widths of the Nikol’skii–Besov classes of periodic functions of many variables in the space of quasicontinuous functions, Ukr. Math. J. 74 (2) (2022) 251–265.
  * [37] A.S. Romanyuk, S.Ya. Yanchenko, Approximation of the classes of periodic functions of one and many variables from the Nikol’skii-Besov and Sobolev spaces, Ukr. Math. J. 74 (6) (2022) 967–980.
  * [38] A.S. Romanyuk, S.Ya. Yanchenko, Estimates for the entropy numbers of the Nikol’skii-Besov classes of functions with mixed smoothness in the space of quasi-continuons functions, Math. Nachr. 296 (6) (2023) 2575–2587.
  * [39] V.N. Temlyakov, Approximation of periodic functions of several variables with bounded mixed difference, Sb. Mat. 41 (1) (1982) 53–66.
  * [40] V.N. Temlyakov, On the approximation of periodic functions of several variables with bounded mixed difference, Dokl. AN SSSR 253 (1980) 544–548; English transl. in Soviet Math. Dokl. 22 (1980).
  * [41] V.N. Temlyakov, Approximation of functions with a bounded mixed difference by trigonometric polynomials, and the widths of some classes of functions, Math. USSR Izv. 20 (1) (1983) 173–187.
  * [42] V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Proc. Steklov Inst. Math. 178 (1989) 1–121.
  * [43] V.N. Temlyakov, Estimates of best bilinear approximations of periodic functions, Proc. Steklov Inst. Math. 4 (1989) 275–293.
  * [44] V.N. Temlyakov, Estimates of the asymptotic characteristics of classes of functions with bounded mixed derivative or difference, Proc. Steklov Inst. Math. 189 (1990) 161–197.
  * [45] V.N. Temlyakov, Approximation of periodic function, Nova Science Publishes, Inc., New York, 1993.
  * [46] V.N. Temlyakov, An inequality for trigonometric polynomials and its application for estimating the entropy numbers, J. Complexity 11 (2) (1995) 293–307.
  * [47] V.N. Temlyakov, An inequality for trigonometric polynomials and its application for estimating the Kolmogorov widths, East J. Approx. 2 (1996) 253–262.
  * [48] V.N. Temlyakov, Multivariate approximation, Cambridge University Press, 2018.
  * [49] V. Temlyakov, T. Ullrich, Approximation of functions with small mixed smoothness in the uniform norm, J. Approx. Theory 277 (2022) 105718.
  * [50] V.M. Tikhomirov, Diameters of sets in function spaces and the theory of best approximations, Russian Math. Surveys 15 (3) (1960) 75–111.
  * [51] R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers, 2004\.


[width=]figures/networkarchitecturev04.png

Figure : The proposed model. (a) Tissue tiling and local feature extraction.
(b) Derivation of contextual patch features using self-attention-based graph
convolution, taking spatial and semantic dependency into account. (c) Joint
prediction of grade and depth using multiscale contextual features. (d) Rank-
order constraint on the attention network.

##  1 Methodology

Each WSI is represented as a bag $b$ of patches $X_{b}=\\{x_{n}\\}_{n=1}^{N}$.
$N$ denotes the number of non-overlapping patches in a WSI, and
$b\in\\{1,2...B\\}$, where $B$ is the number of training samples. The patch-
level labels $y_{n}$ are unknown, and we have access only to the bag label
$Y_{b}\in$ {normal, well, moderate, poor}.

###  Tissue Feature Encoding

Tissue feature extraction consists of local feature extraction using self-
supervised learning followed by contextual feature extraction using a graph
convolution network (GCN).

####  Local feature extraction

We leverage self-supervised learning (SSL) to pre-train the patch feature
extractor using unlabeled patches extracted from WSIs. To capture fine-grained
pathological features (e.g., nuclei details, cell distribution, tumor
microenvironment), we extract $448\times 448$ sized non-overlapping patches at
$20X$ magnification. We use Nest-S [zhang2022nested], a hierarchical
transformer, to extract patch features. We pre-train the feature extractor
using DINO (knowledge distillation-based SSL) [caron2021emerging] because of
its promising performance in MIL-based classification tasks [chen2022scaling].
After pre-training, the transformer network is used as an offline feature
extractor to derive $d$-dimensional feature $f_{n}\in\mathcal{R}^{d}$ for each
patch $x_{n}$, leading to a WSI representation of $\\{f_{n}\\}_{n=1}^{N}$.
Since the pre-trained features $f_{n}$ are agnostic to the downstream task, we
project them into a lower-dimensional space using a multi-layer perceptron
(MLP) with nonlinear activation to get local patch features $h_{n}^{0}$. We
train the MLP along with the graph convolution network and the rank-aware
grade classifier. Thus, the resulting local patch feature $h_{n}^{0}$
potentially captures information specific to grade and de-emphasizes
information irrelevant to the downstream task.

####  Contextual feature extraction

Graph Definition: To extract contextual information capturing spatial and
semantic dependency, we derive an undirected graph from each WSI $G=(V,A)$
with patches as nodes $V$ and adjacency matrix $A\in\mathcal{R}^{N\times N}$,
which represents the connections between the nodes. The edges capture the
pathology-related structure and interdependence among tumor regions via two
types of contextual dependencies incorporated into the adjacency matrix $A$:
(i) $A_{sem}$, which represents non-local dependencies between patches that
may be spatially distant, but are similar in terms of their tissue structure
(grade) in the feature space; and (ii) $A_{sp}$, which captures local
dependencies between a tumor patch and its spatially neighboring patches in
the tumor microenvironment. We only consider edges to the K-nearest neighbors
in both the semantic and spatial space. $A_{sem}$ is defined using pairwise
feature similarity between patches:

$A_{sem}(i,j)=\exp\bigg{(}\frac{-d^{sem}_{ij}}{0.1}\bigg{)};d^{sem}_{ij}=1-\frac{f_{i}\cdot
f_{j}}{||f_{i}||_{2}||f_{j}||_{2}}$ (1)

where $d_{ij}^{sem}$ represents the semantic distance between patches $i,j$
with features $f_{i}$ and $f_{j}$. $j\in[1,N]$ denotes the $K$-nearest
neighbors of the $i^{th}$ patch ($K=4$). We pre-process the derived semantic
graph $A_{sem}$ using personalized PageRank kernel-based graph diffusion
[gasteiger2019diffusion]. This amplifies long-range connections between tumor
regions by generating additional edges beyond 1-hop neighbors in the feature
space. $A_{sp}$ is computed using inverse distance weighing across spatially
K-nearest patches.

$A_{sp}(i,j)=\exp\bigg{(}\frac{-d^{sp}_{ij}}{2}\bigg{)};d^{sp}_{ij}=\sqrt{(s_{i}-s_{j})^{2}+(t_{i}-t_{j})^{2}}$
(2)

where $d_{ij}^{sp}$ represents the spatial distance between patches $i,j$ with
spatial coordinates $(s_{i},t_{i})$ and $(s_{j},t_{j})$, respectively. We use
$K=8$ to connect each patch to its immediate neighboring patches in the
tissue. The adjacency matrix $A$ is the average of the spatial and semantic
components $\bigg{(}A=\frac{A_{sem}+A_{sp}}{2}\bigg{)}$.

Graph Feature Aggregation: We leverage graph convolution with residual mapping
[li2020deepergcn] to derive multiscale contextual patch features
$(h^{1}_{n},h^{2}_{n})$ from the local patch feature $h^{0}_{n}$ and adjacency
matrix $A$. Each convolution layer uses message-passing to propagate feature
information from the 1-hop connected nodes $j$ of a node $i$ ($A_{ij}>0$) and
updates node features via the following operation:

$H^{l+1}=H^{l}+GConv(H^{l},A;W^{l})$ (3)

where $H^{l}=[h^{l}_{1},h^{l}_{2}...,h^{l}_{N}]$ and $W^{l}$ are the trainable
parameters of layer $l$. We use two $GConv(\cdot)$ layers and consider two
design choices for $GConv(\cdot)$.

  1. 1.

Fixed edge weights: We use vanilla graph convolution network
(GCN)[kipf2016semi] with weighted message passing using predefined edge
weights from adjacency matrix $A$:

$GConv(\cdot)=ReLU(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{l}W^{l})$
(4)

where $\tilde{D}=\sum_{j}{\tilde{A}(i,j)}$ is the diagonal matrix of
$\tilde{A}=A+I$. $\tilde{A}$ is used to avoid oversmoothing [kipf2016semi].

  2. 2.

Dynamic edge weights: Alternatively, we can leverage graph attention-based
convolution [shi2020masked] to dynamically define edge weights. This allows us
to dynamically aggregate information from patches connected to a patch based
on their pairwise similarity. We use a single attention head with masking to
aggregate the neighboring patch features:

$GConv(\cdot)=ReLU\bigg{[}softmax\bigg{(}\frac{Q^{l}K^{l}}{\sqrt{d}}+M\bigg{)}H^{l}W^{l}\bigg{]}$
(5)

where $Q^{l}=H^{l}W_{Q}^{l},K^{l}=H^{l}W_{K}^{l}$ and $M$ is the attention
mask ($M_{ij}=0$ if $A_{ij}>0$, else $M_{ij}=-\infty$). $W^{l}_{Q},W^{l}_{K}$,
$W^{l}$ are trainable parameters of layer $l$ and $W^{l}_{Q},W^{l}_{K}$ are
used to learn the edge weights through the dot product $Q^{l}K^{l}$.

###  Multi-task training

Multi-task learning allows us to use additional information from auxiliary
labels (depth) to aid main task prediction (grade). We selected depth as an
auxiliary label because (i) it is easily available from diagnostic reports,
(ii) it is derived from pathology images and reflects tissue structure
surrounding the tumor, and (iii) it is well-correlated with grade (Dunn’s
pairwise p-values are significant at a 10% level). We designed separate
predictors for depth and grade while the graph network is shared between them
to allow feature sharing and to ensure parameter efficiency.

####  Grade Classification

We use attention-based feature aggregation to determine the contribution of
each patch to overall WSI prediction. Attention-based aggregation outperforms
traditional pooling-based methods by up-weighing the most relevant patches. We
leverage 1-hop contextual features to determine the normalized attention
weight for each patch using a two-layer network:

$w_{n}=\frac{\exp[a^{T}\tanh(U\cdot
h^{1}_{n})]}{\sum_{i}\exp[a^{T}\tanh(U\cdot h^{1}_{n})]}$ (6)

where $a\in\mathcal{R}^{d}$ and $U\in\mathcal{R}^{d\times d}$ are learnable
parameters. The attention score is based on the formulation proposed by Ilse
et al. [ilse2018attention]. The overall WSI representation is the attention-
weighted average of normalized 1-hop patch features:
$W^{diff}_{b}=\sum_{n=1}^{N}w_{n}\frac{h^{1}_{n}}{||h^{1}_{n}||}$. The WSI
class-likelihood is computed using a single-layer MLP cosine softmax
classifier $\phi$:

$p_{b,c}=\frac{\exp[-\mathcal{D}(\phi(W^{diff}),z_{c})]}{\sum_{c^{\prime}}\exp[-\mathcal{D}(\phi(W^{diff}),z_{c^{\prime}})]}$
(7)

where $c\in\\{0,1,2,3\\}$ representing {normal, well, moderate, poor},
$\mathcal{D}$ represents cosine distance and $z_{c}$ is the prototype (class
centroid) of grade class $c$ defined as
$z_{c}=\frac{1}{N_{c}}\sum_{n:y_{n}=c}\phi(\frac{h_{n}^{1}}{||h_{n}^{1}||})$.
To counter class imbalance due to a lower proportion of poorly differentiated
cases, we leverage class-balanced sampling [cui2019class] during training. We
use the ground-truth WSI grade label $Y_{b}$ to compute our MIL-based grade
classification loss:

$\mathcal{L}_{grade}=-\sum_{b=1}^{B}\sum_{c}\mathbf{1}(Y_{b}=c)\log(p_{b,c}),$
(8)

####  Ordinal Ranking of Patches

We apply two ranking constraints on the attention network to ensure consistent
ranking of tumor regions: (i) an interclass constraint to impose higher
attention values for worse patches, and (ii) an intraclass constraint to
impose higher attention values for more likely patches within the same class.
Interclass ranking: Pathologists determine the WSI grade based on the grade of
the most severe tumor region by implicitly ranking the different tumor
sections based on their severity. Motivated by this, we propose to enforce a
ranking by using pairwise inequality constraints between patches, such that a
more severe patch is ranked higher by the attention network
($w_{n}^{normal}<w_{n}^{well}<w_{n}^{moderate}<w_{n}^{poor})$. To do so, we
require the grade of each patch, which is unavailable. Therefore, we pseudo
label the patches using a threshold on their class probabilities $p_{n,c}$:

$p_{n,c}=p(y_{n}=c|h^{1}_{n})=\frac{\exp[-\mathcal{D}(\phi(h_{n}^{1}),z_{c})]}{\sum_{c^{\prime}}\exp[-\mathcal{D}(\phi(h_{n}^{1}),z_{c^{\prime}})]}$
(9)

We set $y^{pseudo}_{n}$ = $c$ if $p_{n,c}$ $>$ $0.6$, $c$ is the grade class.
We derive a set of pairs $(i,j)$ of pseudo-labeled patches belonging to two
adjacent classes, $Z$ =
$\\{(i,j);y^{pseudo}_{i}=c,y^{pseudo}_{j}=c+1,\text{forsome}c\in\\{0,1,2\\}\\}$
and impose a soft ordinal constraint on their attention weights
($w_{i}<w_{j}$) using the pairwise ranking loss:

$\mathcal{L}_{inter}=\sum_{i,j\in Z}\log[1+\exp(w_{i}-w_{j})]$ (10)

Intraclass ranking: In addition, we impose intraclass ranking on the patches
to ensure that the most confident patches (with higher $p_{n,c}$) within a
particular grade class are weighted higher during feature aggregation. We
focus on the patches with the highest attention weights or class probability,
i.e., $\mathcal{S}=\\{n;n\in TopK(w_{n})\cup TopK(p_{n,c})\\}$ for $K=50$.
Next, we derive a set of pairs $(i,j)$ using their class probabilities
$\tilde{Z}$ =
$\\{(i,j);p_{i,c}<p_{j,c}-\delta,\text{for}i,j\in\mathcal{S},c\in\\{0,1,2,3\\}\\}$,
where $\delta=0.1$ is used to limit the number of pairs due to computational
constraints. We impose a pairwise ranking loss on the attention network for
patches in $\tilde{Z}$:

$\mathcal{L}_{intra}=\sum_{i,j\in\tilde{Z}}\log[1+\exp{(w_{i}-w_{j})}]$ (11)

For a pair of patches $(i,j)$, this loss enforces that if $p_{j,c}>p_{i,c}$,
then attention weight $w_{j}>w_{i}$.

####  Depth Prediction

Depth values are continuous and depend on the global tissue structure. Since
depth is continuous, we formulate depth prediction as a regression task. The
depth predictor uses a single-layer MLP regressor $W_{reg}$ and a two-layer
attention network. We use 2-hop patch features $h^{2}_{n}$ for both regression
and attention networks since it captures more contextual information, which is
useful for depth prediction. The WSI-level feature for depth prediction is
$H_{b}^{depth}=\sum_{n=1}^{N}w^{d}_{n}\cdot h^{2}_{n}$, where $w^{d}_{n}$ is
the attention weight computed from $h^{2}_{n}$ by using an equation similar to
equation 6. The depth predictor is trained using the robust Geman-McClure loss
[barron2019general] to reduce sensitivity to large errors from outlier depth
values ($>10mm$):

$\mathcal{L}_{depth}=\frac{2(\hat{Y}_{b}^{depth}-Y_{b}^{depth})^{2}/c^{2}}{2(\hat{Y}_{b}^{depth}-Y_{b}^{depth})^{2}/c^{2}+4}$
(12)

where $\hat{Y}_{b}^{depth}=W_{reg}\cdot(H^{depth}_{b})^{T}$ and $c=2$ is the
scale parameter.
Overall Loss: The overall loss combines the grade classification loss, depth
prediction loss, and the interclass and intraclass attention ranking losses.
To balance the losses, we used the weighing factors $\lambda_{0}$,
$\lambda_{1}$ and $\lambda_{2}$, which are determined using hyperparameter
tuning:

$\mathcal{L}_{total}=\mathcal{L}_{grade}+\lambda_{0}\mathcal{L}_{depth}+\lambda_{1}\mathcal{L}_{inter}+\lambda_{2}\mathcal{L}_{intra}\\\
$ (13)