00footnotetext: ; Institute of Mathematics, University of Rzeszów, Poland

# Some exact constants for bilateral approximations in quasi-normed groups

Oleh Lopushansky

###### Abstract

We establish inverse and direct theorems on best approximations in quasi-
normed Abelian groups in the form of bilateral Bernstein-Jackson inequalities
with exact constants. Using integral representations for quasi-norms of
functions $f$ in Lebesgue’s spaces by decreasing rearrangements $f^{*}$ with
the help of approximation $E$-functionals, error estimates are found. Examples
of numerical calculations, spectral approximations of self-adjoint operators
obtained by obtained estimates are given.

###### keywords:

Bernstein-Jackson inequalities, quasi-normed Abelian groups, exact
approximation constants

###### pacs:

[

MSC Classification]42B35, 41A44, 41A17,46B70

##  1 Introduction and main results

This article is devoted to the proof of a bilateral version of Bernstein-
Jackson inequalities for quasi-normed Abelian groups. The case of similar
inequalities on Gaussian Hilbert spaces of random variables was analyzed in
the previous publication Lopushansky2023 . In the studied case, we use the
approximation scale

$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta},\quad
s+1=1/\vartheta,\quad\tau=\vartheta q$

of compatible couple of quasi-normed Abelian groups with ${0<\vartheta<1}$ and
${0<q\leq\infty}$, defined by the Lions-Peetre method of real interpolation.
It should be noted that in partial cases this scale coincides with the known
scale of Besov spaces (see (bergh76, , Thm 7.2.4), DeVore1988 ; DeVore1998
,(Nikolski75, , Thm 2.5.4)).

We also analyze the problem of accurate error estimates in quasi-normed spaces
$L^{p}_{\mu}=L^{p}_{\mu}(\mathfrak{A})$ $(0\leq p\leq\infty)$ of
$\mathfrak{A}$-valued functions ${f\colon X\to\mathfrak{A}}$ on a measure
space $(X,\mu)$. Using the decreasing rearrangement method (see e.g. Burchard
; Grafakos2014 ), we reduce this problem to the simpler case for uniquely
defined decreasing functions $f^{*}$ on $(0,\infty)$.

The scale of quasi-normed spaces
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ endowed with the
quasinorm ${\|\cdot\|_{\mathcal{B}_{\tau}^{s}}}$ is determined by the best
approximation $E$-functional

$E(t,a\mathchar
24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\left\\{|a-a_{0}|_{1}\colon|a_{0}|_{0}<t\right\\},\quad
a\in{\mathfrak{A}_{1}}$

which fully characterizes the accuracy of error estimates (see e.g. bergh76 ;
Maligranda1991 ; PeetreSparr1972 ). In Theorem 1 we establish the bilateral
version of Bernstein-Jackson inequalities for quasi-normed groups with the
exact constants $c_{s,\tau}$ depend only on basis parameters
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$

$\displaystyle t^{s}E(t,a)\leq{c_{s,\tau}}|a|_{\mathcal{B}_{\tau}^{s}}\leq
2^{1/2}|a|^{s}_{0}|a|_{1},\quad
c_{s,\tau}=\bigg{[}\frac{s}{\tau(s+1)^{2}}\bigg{]}^{1/\tau}$

for all $a\in\mathfrak{A}_{0}\bigcap\mathfrak{A}_{1}$ and $t,\tau>0$. The left
inequality allows the unique extension to the whole approximation spaces
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ in form of the
Jackson-type inequality

$E(t,a)\leq{c_{s,\tau}t^{-s}}\,|a|_{\mathcal{B}_{\tau}^{s}}\quad\text{for
all}\quad{a\in\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})}.$

In Section 3 these results are applied to the quasi-normed Abelian groups
$L^{p}_{\mu}=L^{p}_{\mu}(\mathfrak{A})$ with $0\leq p\leq\infty$ and a
positive Radon measure $\mu$ on a measure space $X$ of measurable functions
${f\colon X\to\mathfrak{A}}$ endowed with the $\kappa$-norm

$\|f\|_{p}=\left\\{\begin{array}[]{ll}\displaystyle\Big{(}\int|f(x)|^{p}\,\mu(dx)\Big{)}^{1/p}&\hbox{if
}0<p<\infty\\\ \mathop{\rm ess\,sup}_{x\in X}|f(x)|&\hbox{if
}p=\infty\\\\[4.30554pt]
\mu\left(\operatorname{\operatorname{supp}}f\right)&\hbox{if
}p=0,\end{array}\right.$

where $\operatorname{\operatorname{supp}}f\subset X$ is a measurable subset
such that $f\mid_{X\setminus\operatorname{\operatorname{supp}}f}=0$ and
${f\neq 0}$ almost everywhere on $\operatorname{\operatorname{supp}}f$ with
respect to the measure $\mu$. In Theorem 7 the equality

$f^{*}(t)=E\big{(}t,f\mathchar
24635\relax\;L_{\mu}^{0},L_{\mu}^{\infty}\big{)},\quad
f\in{L_{\mu}^{\infty}},$

where $f^{*}$ is the decreasing rearrangement of $\mathfrak{A}$-valued
functions $f\in L^{p}_{\mu}$ and, as a consequence, the following equalities

$\|f\|_{\tau/s}=\left\\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{s}f^{*}(t)\right]^{\tau}\frac{dt}{t}\right)^{1/\tau}&\hbox{if
}\tau<\infty\\\\[8.61108pt] \sup\limits_{0<t<\infty}t^{s}f^{*}(t)&\hbox{if
}\tau=\infty,\end{array}\right.$

are established. We also compute exact constants in the Jackson-type
inequalities

$f^{*}(t)\leq t^{s}2^{(s+1)/2}c_{s,\tau}\|f\|_{\tau/s}\quad\text{for
all}\quad{f\in L^{\tau/s}}$

that estimate measurable errors for decreasing rearrangements $f^{*}$, as well
as the bilateral Bernstein-Jackson inequalities

$\displaystyle t^{-s}2^{-(s+1)/2}f^{*}(t)\leq
c_{s,\tau}\|f\|_{\tau/s}\leq\|f\|^{s}_{0}\|f\|_{\infty}\quad\text{for
all}\quad{f\in L^{0}\cap L^{\infty}},\quad\text{where}$ $\displaystyle
2^{-1/2\theta}C_{\theta,q}=\left\\{\begin{array}[]{cl}c_{s,\tau}&\hbox{if
}\tau<\infty\\\ 1&\hbox{if }\tau=\infty.\end{array}\right.$ (3)

In particular, the following inequality holds,

$f^{*}(t)\leq\frac{2}{\pi t}\,\int|f|\,\mu(dx)\quad\text{for all}\quad f\in
L^{1}_{\mu}.$

The numerical algorithm resulting from the Theorem 7 was carried out on the
example of inverse Gaussian distribution. Examples of applications for
accurate estimates of spectral approximations of self-adjoint operators are
given in Section 4.

Note that basic notations used in this work can be found in bergh76 ;
Triebel78 .

##  2 Best approximation scales of quasi-normed Abelian groups

In what follows, we study the $\kappa$-normed Abelian groups
$(\mathfrak{A},{|\cdot|})$ relative to an operation $"+"$, where $\kappa$-norm
${|\cdot|}$ with the constant $\kappa\geq 1$ is determined by the assumptions
(see e.g. bergh76 ; Maligranda1991 ):

> $|0|=0$ and $|a|>0$ for all nonzero $a\in\mathfrak{A}$,
> $|a|=|-a|$ for all $a\in\mathfrak{A}$,
> $|a+b|\leq\kappa(|a|+|b|)$ with $\kappa$ independent on $a,b\in\mathfrak{A}$

As is known, each $\kappa$-norm $|\cdot|$ can be replaced by an equivalent
$1$-norm $|\cdot|^{\prime}$ such that $|\cdot|^{\prime}\leq|\cdot|^{\rho}\leq
2|\cdot|^{\prime}$ by taking $(2\kappa)^{\rho}=2$ with a suitable $\rho>0$
(see e.g. PeetreSparr1972 ). We will not assume the completeness of groups.

Given a compatible couple $(\mathfrak{A}_{0},\mathfrak{A}_{1})$ of
$\kappa_{\imath}$-normed groups $(\mathfrak{A}_{\imath},{|\cdot|_{\imath}})$
with $\imath=\\{0,1\\}$ and $a=a_{0}+a_{1}$ in the sum
${\mathfrak{A}_{0}+\mathfrak{A}_{1}}$ such that
$a_{\imath}\in\mathfrak{A}_{\imath}$ we define the best approximation
$E$-functional $E(t,a\mathchar
24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})$ with
${a\in\mathfrak{A}_{0}+\mathfrak{A}_{1}}$ and ${t>0}$ in the form (see
(bergh76, , no.7))

$E(t,a)=E(t,a\mathchar
24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\left\\{|a-a_{0}|_{1}\colon|a_{0}|_{0}<t\right\\},\quad
a\in{\mathfrak{A}_{1}}.$ (4)

######  Definition 1.

For any $\tau=q\vartheta$ and $s+1=1/\vartheta$ with ${0<\vartheta<1}$ and
${0<q\leq\infty}$ the $\kappa$-normed best approximation scales of Abelian
groups are defined to be

$\begin{split}\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})&=\left\\{a\in\mathfrak{A}_{0}+\mathfrak{A}_{1}\colon|a|_{\mathcal{B}_{\tau}^{s}}<\infty\right\\},\\\
|a|_{\mathcal{B}_{\tau}^{s}}&=\left\\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{s}E(t,a)\right]^{\tau}\frac{dt}{t}\right)^{1/\tau}&\\!\\!\\!\\!\\!\\!\hbox{if
}\tau<\infty\\\\[8.61108pt]
\sup\limits_{0<t<\infty}t^{s}E(t,a)&\\!\\!\\!\\!\\!\\!\hbox{if
}\tau=\infty,\end{array}\right.\end{split}$ (5)

where (by (bergh76, , Lemma 7.1.6))
$\kappa=2\kappa_{1}\max(\kappa_{0}^{s},\kappa_{1}^{s})\max(1,2^{-1/q^{\prime}})\max(1,2^{s-1})$,
$(1/q^{\prime}=1-1/q).$

Given a compatible couple $(\mathfrak{A}_{0},\mathfrak{A}_{1})$, we also
define the quadratic functional of Lions-Peetre’s type (see e.g. McLean2000 ;
PeetreSparr1972 )

$K_{2}(t,a)=K_{2}(t,a\mathchar
24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\limits_{a=a_{0}+a_{1}}\left(|a_{0}|_{0}^{2}+t^{2}|a_{1}|_{1}^{2}\right)^{1/2},\quad
t>0.$

We will use the quadratically modified real interpolation method. Define the
interpolation Abelian group of Lions-Peetre’s type
$\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}$ endowed with
the quasinorm
${\|\cdot\|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}}$,

$\displaystyle K_{\theta,q}\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)$
$\displaystyle=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}=\big{\\{}a\in\mathfrak{A}_{0}+\mathfrak{A}_{1}\big{\\}},$
$\displaystyle|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle{\bigg{(}\int_{0}^{\infty}\left[t^{-\vartheta}K_{2}(t,a)\right]^{q}\frac{dt}{t}\bigg{)}^{1/q}}&\\!\\!\\!\\!\hbox{if
}q<\infty\\\\[8.61108pt]
\sup\limits_{0<t<\infty}t^{-\vartheta}K_{2}(t,a)&\\!\\!\\!\\!\hbox{if
}q=\infty.\end{array}\right.$

Following DL19 ; Lopushansky2023 , we extend the use of approximation
constants $c_{s,\tau}$ to the case of arbitrary quasi-normed Abelian groups
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ described in the
classic works PeetreSparr1972 ; pietsch1981 , where

$c_{s,\tau}:=\left\\{\begin{array}[]{cl}\displaystyle\left[\frac{s}{\tau(s+1)^{2}}\right]^{1/\tau}&\hbox{if
}\tau<\infty\\\ \displaystyle 1&\hbox{if }\tau=\infty\end{array}\right.$ (6)

is determined by the normalization factor $N_{\vartheta,q}$ from (bergh76, ,
Thm 3.4.1) to be

$c_{s,\tau}=N^{1/q}_{\vartheta,q}\left(\vartheta
q^{2}\right)^{-1/q\vartheta},\quad
N_{\vartheta,q}:=[q\vartheta(1-\vartheta)]^{1/q}.$ (7)

The following theorem establishes the bilateral form of approximation
inequalities with exact constants for the case of quasi-normed Abelian groups.

######  Theorem 1.

The bilateral Bernstein-Jackson inequalities with the exact constant
${c}_{s,\tau}$

$\displaystyle t^{s}E(t,a)$
$\displaystyle\leq{c}_{s,\tau}|a|_{\mathcal{B}_{\tau}^{s}}\leq
2^{1/2}|a|^{s}_{0}|a|_{1}\quad\text{for all}\quad
a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1}$ (8)

hold. There is a unique extension of Jackson’s inequality on the whole
approximation scale

$\displaystyle E(t,a)$ $\displaystyle\leq
t^{-s}{c}_{s,\tau}|a|_{\mathcal{B}_{\tau}^{s}}\quad\text{for
all}\quad{a\in\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})},\ \
r>0.$ (9)

Proof: Let ${0<q<\infty}$ and
$\alpha=|a|_{\mathfrak{A}_{1}}/|a|_{\mathfrak{A}_{0}}$ with a nonzero
$a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1}$. Since

$\displaystyle K_{2}(t,a)^{2}$
$\displaystyle=\inf\limits_{a=a_{0}+a_{1}}\left(|a_{0}|_{0}^{2}+t^{2}|a_{1}|_{1}^{2}\right)\leq|a|^{2}_{0}\min(1,\alpha^{2}t^{2})={\min\left(|a|^{2}_{0},t^{2}|a|^{2}_{1}\right)}$

or otherwise $K_{2}(t,a)\leq{|a|_{0}\min(1,\alpha
t)}={\min\left(|a|_{0},t|a|_{1}\right)},$ we get the inequality

$|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\leq|a|_{1}^{q}\int_{0}^{\alpha}t^{-1+q(1-\vartheta)}dt+|a|^{q}_{0}\int_{\alpha}^{\infty}t^{-1-\vartheta
q}dt.$

Calculating the integrals and using that $\alpha=|a|_{1}/|a|_{0}$, we obtain

$\displaystyle|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$
$\displaystyle\leq\frac{\alpha^{q(1-\vartheta)}}{q(1-\vartheta)}|a|_{1}^{q}+\frac{\alpha^{-\vartheta
q}}{\vartheta
q}|a|^{q}_{0}=\frac{1}{q\vartheta(1-\vartheta)}\left(|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}\right)^{q}.$

Let it be now $q=\infty$. Then the inequality

$t^{-\vartheta}K_{2}(t,a)\leq{\min\left(t^{-\vartheta}|a|_{0},t^{1-\vartheta}|a|_{1}\right)}$

holds. Taking in this case $t=|a|_{0}/|a|_{1}$, we obtain that
$t^{-\vartheta}K_{2}(t,a)\leq|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}.$
Combining previous inequalities and taking into account (7), we have

$\begin{split}\|a\|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}&\leq\left\\{\begin{array}[]{ll}N_{\vartheta,q}^{-1}|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if
}q<\infty\\\\[6.45831pt] |a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if
}q=\infty\end{array}\right.,\quad
a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1}.\end{split}$ (10)

For further considerations, we need the functional

$K_{\infty}(v,a):=\inf\limits_{a=a_{0}+a_{1}}\max\big{(}|a_{0}|_{\mathfrak{A}_{0}},v|a_{1}|_{\mathfrak{A}_{1}}\big{)}.$

Now we will use the known properties of the considered functionals that

$\displaystyle v^{-\theta}K_{\infty}(v,f)\to 0\quad\text{as \ }{v\to 0}\text{
or }{v\to\infty}$ $\displaystyle{t^{-1+1/\theta}E(t,f)\to 0}\quad\text{as \
}{v\to 0}\text{ or }{v\to\infty}$

(see (bergh76, , Thm 7.1.7)). As a result, we get

$\displaystyle\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v}$
$\displaystyle=-\frac{1}{\vartheta
q}\int_{0}^{\infty}K_{\infty}(v,a)^{q}dv^{-\vartheta q}=\frac{1}{\vartheta
q}\int_{0}^{\infty}v^{-\vartheta q}dK_{\infty}(v,a)^{q}.$

On the other hand, integrating by parts with the change $v=t/E(t,a)$, we get
for $s+1=1/\vartheta$ that

$\displaystyle\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v}$
$\displaystyle=\frac{1}{\vartheta
q}\int_{0}^{\infty}\left(t/E(t,a)\right)^{-\vartheta
q}dt^{q}=\frac{1}{\vartheta
q^{2}}\int_{0}^{\infty}\left(t^{s}E(t,a)\right)^{\vartheta q}\frac{dt}{t}.$

From the definition $K_{\infty}$ and $K_{2}$, it follows that

$K_{\infty}(t,a)\leq K_{2}(t,a)\leq 2^{1/2}K_{\infty}(t,a)$ (11)

(see (PeetreSparr1972, , Remark 3.1)). Now, taking into account that

$\frac{1}{\vartheta q^{2}}\|a\|^{\vartheta
q}_{\mathcal{B}_{\tau}^{s}}=\frac{1}{\vartheta
q^{2}}\int_{0}^{\infty}\left(t^{s}E(t,a)\right)^{\vartheta
q}\frac{dt}{t}=\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v},$

according to the left inequality from (11), we obtain

$\frac{1}{\vartheta q^{2}}|a|^{\vartheta
q}_{\mathcal{B}_{\tau}^{s}}\leq\int_{0}^{\infty}\left(v^{-\vartheta}K_{2}(v,a)\right)^{q}\frac{dv}{v}=|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}.$
(12)

By using the right inequality from (11), we have

$\displaystyle|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}=\int_{0}^{\infty}\left(v^{-\vartheta}K_{2}(v,a)\right)^{q}\frac{dv}{v}\leq
2^{q/2}\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v}$
$\displaystyle=\frac{2^{q/2}}{\vartheta
q^{2}}\int_{0}^{\infty}\left(t^{s}E(t,a)\right)^{\vartheta
q}\frac{dt}{t}=\frac{2^{q/2}}{\vartheta q^{2}}|a|^{\vartheta
q}_{\mathcal{B}_{\tau}^{s}}.$

As a result, by virtue of (12) we have the inequalities

$|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\leq\frac{2^{q/2}}{\vartheta
q^{2}}|a|^{\vartheta q}_{\mathcal{B}_{\tau}^{s}}\leq
2^{q/2}|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\quad\text{with}\quad\tau=\vartheta
q.$ (13)

That gives the following isomorphism with equivalent quasinorms

$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta}.$
(14)

By (bergh76, , Lemma 7.1.2 and Thm 7.1.1) for each ${v>0}$ there is ${t>0}$
such that

$(t^{s}E(t,a))^{\vartheta}\leq
v^{-\vartheta}K_{\infty}(v,a)\leq\left(t^{s}E(t-0,a)\right)^{\vartheta}.$ (15)

Since
$|a|^{\vartheta}_{\mathcal{B}_{\infty}^{s}}\leq|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$
for $q=\infty$, the inequalities (15) for any ${v>0}$ yield

$\displaystyle v^{-\vartheta}K_{2}(v,a)$ $\displaystyle\leq
v^{-\vartheta}2^{1/2}K_{\infty}(v,a)\leq
2^{1/2}\big{(}t^{s}E(t-0,a)\big{)}^{\vartheta}$ $\displaystyle\leq
2^{1/2}\Big{(}\sup_{t>0}\,t^{s}E(t,a)\Big{)}^{\vartheta}=2^{1/2}|a|^{\vartheta}_{\mathcal{B}_{\infty}^{s}}.$

As a result, $|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\leq
2^{1/2}|a|^{\vartheta}_{\mathcal{B}_{\infty}^{s}}$ and the isomorphism (14)
holds for $q=\infty$.

Prove the Bernstein inequality (33). Combining (10) and (12), we obtain

$|a|^{\vartheta}_{\mathcal{B}_{\tau}^{s}}\leq\left\\{\begin{array}[]{ll}\displaystyle\left[\frac{q}{1-\vartheta}\right]^{1/q}|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if
}q<\infty\\\\[8.61108pt] |a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if
}q=\infty.\end{array}\right.$ (16)

Thus, the second inequality (33) we get by setting $s+1=1/\vartheta$ and
$\tau=\vartheta q$.

Prove the inequality (32). Integrating
${\min(1,v/t)K_{2}(t,a)\leq{K}_{2}(v,a)}$, we obtain

$\int_{0}^{\infty}\left(v^{-\vartheta}\min(1,v/t)\right)^{q}\frac{dv}{v}K_{2}(t,a)^{q}\leq\int_{0}^{\infty}\left(v^{-\vartheta}K_{2}(v,a)\right)^{q}\frac{dv}{v}=|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}^{q}.$

Since the left integral can be rewritten as the following sum

$\begin{split}\int_{0}^{\infty}\left(v^{-\vartheta}\min(1,v/t)\right)^{q}\frac{dv}{v}&=\int_{0}^{t}v^{(1-\vartheta)q-1}t^{-q}dv+\int_{t}^{\infty}v^{-\vartheta
q-1}dv=\frac{1}{t^{\vartheta}N_{\vartheta,q}^{1/q}},\end{split}$

we obtain the inequality

$\begin{split}\bigg{(}\int_{0}^{t}\frac{v^{(1-\vartheta)q-1}}{t^{q}}dv+\int_{t}^{\infty}v^{-\vartheta
q-1}dv\bigg{)}^{1/q}K_{2}(t,a)&=\frac{K_{2}(t,a)}{t^{\vartheta}N_{\vartheta,q}}\leq|x|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}.\end{split}$

As a result, $K_{2}(t,a)\leq
t^{\vartheta}N_{\vartheta,q}|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$.
Hence, taking into account (11) and (15), we get

$v^{1-\vartheta}E(v,a)^{\vartheta}\leq t^{-\vartheta}K_{\infty}(t,a)\leq
N_{\vartheta,q}|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}.$

Applying (13), we obtain

$v^{1-\vartheta}E(v,a)^{\vartheta}\leq{2^{1/2}}N_{\vartheta,q}|a|^{\vartheta}_{\mathcal{B}_{\tau}^{s}}.$

Taking into account that $\vartheta=1/(s+1)=q/\tau,$ we obtain the inequality
(32) for all ${0<q<\infty}$.

In the case $q=\infty$, we have

$t^{s}E(t,a)\leq\sup_{t>0}\,t^{s}E(t,a)=|a|_{\mathcal{B}_{\infty}^{s}}$

for all ${a\in\mathcal{B}_{\infty}^{s}}$. Thus, the inequality (32) holds for
both cases.

Finally note that Bernstein-Jackson inequalities are achieved at
$\tau=\infty$, so they are sharp.

######  Corollary 2.

If the quasi-normed Abelian groups $\mathfrak{A}_{0}$, $\mathfrak{A}_{1}$ are
compatible then the following isomorphism with equivalent quasi-norms holds,

$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta},\quad
s+1=1/\vartheta,\quad\tau=\vartheta q.$ (17)

If compatible groups $\mathfrak{A}_{0}$ and $\mathfrak{A}_{1}$ are complete,
the approximation group
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ is complete.

Proof: The isomorphism (17) immediately follows from (14) and Theorem 1. Note
that if both compatible groups $\mathfrak{A}_{0}$ and $\mathfrak{A}_{1}$ are
complete, the interpolation group
$\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}$ is complete
(bergh76, , Thm 3.4.2 & Lemma 3.10.2). By Theorem 1
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ is complete.

######  Corollary 3.

Let $(\mathfrak{A}_{0},\mathfrak{A}_{1})$,
$(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})$ be compatible. If
$T\colon\mathfrak{A}_{\imath}\to\mathfrak{A}^{\prime}_{\imath}$ $(\imath=0,1)$
are quasi-Abelian mappings, i.e. ${T(a_{0}+a_{1})}={b_{0}+b_{1}}$ and
$|Ta_{\imath}|_{\mathfrak{A}_{\imath}}\leq\kappa_{\imath}|b_{\imath}|_{\mathfrak{A}^{\prime}_{\imath}}$
with constants $\kappa_{\imath}$ (bergh76, , p.81), then there exists a
constant $C>0$ such that

$T\colon\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})\to\mathcal{B}_{\tau}^{s}(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})\quad\text{and}\quad|Ta|_{\mathcal{B}_{\tau}^{s}(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})}\leq
C|a|_{\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})}.$

Proof: It follows from (13), (14) and the boundness of

$T\colon(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}\to(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})_{\vartheta,q},$

where $C\leq 4^{1/\vartheta}(\vartheta q^{2})^{-1/\vartheta
q}\kappa_{0}^{s}\,\kappa_{1}$ that gives the required inequality.

######  Corollary 4.

Let $0\leq\vartheta_{0}<\vartheta_{1}\leq 1$, ${0<q\leq\infty}$ and
$\vartheta=(1-\eta)\vartheta_{0}+\eta\vartheta_{1}$ ${(0<\eta<1)}$. If the
quasi-normed Abelian groups $(\mathfrak{A}_{0},\mathfrak{A}_{1})$,
$(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})$ are compatible and the
inclusions

$(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta_{\imath},q}\subset\mathfrak{A}^{\prime}_{\imath}\subset(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta_{\imath},\infty},\quad(\imath=0,1)$

are valid, the following isomorphism with equivalent quasinorms holds,

$\mathcal{B}_{\vartheta
q}^{(1-\vartheta)/\vartheta}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\mathcal{B}_{\eta
q}^{(1-\vartheta)/\vartheta}(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1}).$
(18)

Proof: Using (14) and the known equality
$\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}=\left(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1}\right)_{\eta,q}$
PeetreSparr1972 , (Komatsu1981, , Theorem 3) in the case ${r=1}$, we get the
reiteration identity (18) with accuracy to quasinorm equivalency.

######  Corollary 5.

The following continuous embedding hold

$\mathcal{B}_{\vartheta
q}^{(1-\vartheta)/\vartheta}(\mathfrak{A}_{0},\mathfrak{A}_{1})\looparrowright\mathcal{B}_{\vartheta
p}^{(1-\vartheta)/\vartheta}(\mathfrak{A}_{0},\mathfrak{A}_{1})\quad\text{with}\quad
q<p.$

Proof: It follows from (14), since for arbitrary $q<p$ the following
continuous embedding
$(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}\looparrowright(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,p}$
is true by virtue of (Komatsu1981, , Lemma p. 385).

######  Corollary 6.

If $\vartheta_{0}<\vartheta_{1}$ then the embedding
$\mathfrak{A}_{1}\subset\mathfrak{A}_{0}$ yields

$\mathcal{B}_{\vartheta_{1}q}^{(1-\vartheta_{1})/\vartheta_{1}}(\mathfrak{A}_{0},\mathfrak{A}_{1})\subset\mathcal{B}_{\vartheta_{0}q}^{(1-\vartheta_{0})/\vartheta_{0}}(\mathfrak{A}_{0},\mathfrak{A}_{1}).$

Proof: It follows from Corollary 4 and (bergh76, , Theorem 3.4.1(d)).

##  3 Exact constants in bilateral error estimates

Let ${(\mathfrak{A},|\cdot|)}$ be a $\kappa$-normed Abelian group. Consider a
quasi-normed space $L^{p}_{\mu}=L^{p}_{\mu}(\mathfrak{A})$ with a positive
Radon measure $\mu$ on a measure space $(X,\mu)$ of measurable functions
${f\colon X\to\mathfrak{A}}$ endowed with the $\kappa$-norm

$\|f\|_{p}=\left\\{\begin{array}[]{ll}\displaystyle\Big{(}\int|f(x)|^{p}\,\mu(dx)\Big{)}^{1/p}&\hbox{if
}0<p<\infty\\\ \mathop{\rm ess\,sup}_{x\in X}|f(x)|&\hbox{if
}p=\infty\\\\[4.30554pt]
\mu\left(\operatorname{\operatorname{supp}}f\right)&\hbox{if
}p=0,\end{array}\right.$

where $\operatorname{\operatorname{supp}}f\subset X$ is a measurable subset
such that $f\mid_{X\setminus\operatorname{\operatorname{supp}}f}=0$ and
${f\neq 0}$ almost everywhere on $\operatorname{\operatorname{supp}}f$ with
respect to the measure $\mu$.

For a $\mu$-measurable function ${f\colon X\to\mathfrak{A}}$, we define the
distribution function

$m(\sigma,f)=\mu\left\\{x\in X\colon{|f(x)|>\sigma}\right\\}.$

Denote by $f^{*}$ the decreasing rearrangement of $f$, where

$f^{*}(t):=\inf\left\\{\sigma>0\colon m(\sigma,f_{\sigma})\leq t\right\\}$

with $f_{\sigma}(x)=f(x)$ if $|f(x)|>\sigma$ and $f_{\sigma}(x)=0$ otherwise
(for details see (bergh76, , no 1.3)). The decreasing rearrangement $f^{*}$
is nonnegative nonincreasing continuous on the right function of $\sigma$ on
$(0,\infty)$ which is equimeasurable with $f$ in the sense that

$m(\sigma,f)=m(\sigma,f^{*})\quad\text{for all}\quad\sigma\geq 0.$

In other words, the decreasing rearrangement of a function $f$ is a
generalized inverse of its distribution function in the sense that if
$m(\sigma,f)$ is one-to-one then $f^{*}$ is simply the inverse of
$m(\sigma,f)$.

Since $f$ is rearrangeable, $\left\\{\sigma>0\colon m(\sigma,f_{\sigma})\leq
t\right\\}$ is nonempty for $t>0$ thus $f^{*}$ is finite on its domain.

In the case where $\mu(X)<\infty$, we consider $f^{*}$ as a function on
$(0,\mu(X))$, since $f^{*}(t)=0$ for all $t>\mu(X)$. The set
$\left\\{\sigma>0\colon m(\sigma,f_{\sigma})=0\right\\}$ can be empty. If $f$
is bounded then $f^{*}\to\|f\|_{\infty}$ as $t\to 0_{+}$, otherwise, $f^{*}$
is unbounded at the origin.

As is also know (see e.g. ONeilWeiss1963 ) for $f\in L^{p}_{\mu}$ with $p\geq
1$ that

$\int|f|^{p}\,\mu(dx)=p\int_{0}^{\infty}\sigma^{p-1}m(\sigma,f)\,d\sigma,\quad\|f\|_{p}^{p}=\int_{0}^{\infty}f^{*}(t)^{p}\,dt.$

The couple $\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$ is compatible in the
interpolation theory sense (see e.g. (bergh76, , Lemma 3.10.3) or
(Komatsu1981, , no 1)). Given elements $f={f_{0}+f_{\infty}}$ in the
algebraic sum ${L^{0}_{\mu}+L^{\infty}_{\mu}}$, we define the best
approximation $E$-functional $E\left(t,f\mathchar
24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\right)$ with ${t>0}$ as (see e.g.
(bergh76, , no 7))

$\displaystyle E(t,f)$ $\displaystyle=E\big{(}t,f\mathchar
24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=\inf\left\\{\|f-f_{0}\|_{\infty}\colon\|f_{0}\|_{0}<t\right\\},\quad
f\in{L^{\infty}_{\mu}}.$

In what follows , we use the equivalent parameters ${0<\vartheta<1},\
{0<q\leq\infty}$, where $s+1=1/\vartheta$ and $\tau=\vartheta q$. Following
Lopushansky2023 , the appropriate system of exact constants $C_{\theta,q}$ is
defined by the normalization factor

$N_{\theta,q}=\left(\int_{0}^{\infty}|t^{-\theta}\mathcal{N}(t)|^{q}\frac{dt}{t}\right)^{-1/q}\quad\text{with}\quad\mathcal{N}(t)=\frac{t}{\sqrt{1+t^{2}}}$

in the known interpolation Lions-Peetre method to be

$\begin{split}C_{\theta,q}&=\left\\{\begin{array}[]{cl}\displaystyle\frac{2^{1/2\theta}}{(q^{2}\theta)^{1/q\theta}}N_{\theta,q}^{1/\theta}&\hbox{if
}q<\infty,\\\\[8.61108pt]
\displaystyle\left(\frac{\sin\pi\theta}{\pi\theta}\right)^{1/2\theta}&\hbox{if
}q=2,\\\\[8.61108pt] \displaystyle 2^{1/2\theta}&\hbox{if
}q=\infty.\end{array}\right.\end{split}$ (19)

######  Remark 1.

For this equivalent parameter system the sharp constant $c_{s,\tau}$ takes an
equivalent form

$2^{-1/2\theta}C_{\theta,q}=\left\\{\begin{array}[]{cl}c_{s,\tau}&\hbox{if
}\tau<\infty\\\ 1&\hbox{if }\tau=\infty\end{array}\right.,$

where the following dependencies between indexes are performed,

$2^{-1/2\theta}C_{\theta,q}=\left[(1-\theta)/q\right]^{1/q\theta}=(\theta
q^{2})^{-1/q\theta}N_{\theta,q}^{1/\vartheta}.$

A graphical image of the normalized constant sharp constant $c_{s,\tau}$ in
variables $s,\tau$ shown on Fig. 1.

Figure 1: 3D-graph of $c_{s,\tau}=(s/\tau)^{1/\tau}(s+1)^{-2/\tau}$ for
${s,\tau>0}$

######  Theorem 7.

(a) For any ${0<\vartheta<1}$ and ${0<q\leq\infty}$ the equality

$f^{*}(t)=E\big{(}t,f\mathchar
24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)},\quad f\in{L^{\infty}_{\mu}}$
(20)

and, as a consequence, the following equalities hold,

$\|f\|_{q\theta^{2}/(1-\theta)}=\left\\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{-1+1/\theta}f^{*}(t)\right]^{q\theta}\frac{dt}{t}\right)^{1/q\theta}&\hbox{if
}q<\infty\\\\[8.61108pt]
\sup\limits_{0<t<\infty}t^{-1+1/\theta}f^{*}(t)&\hbox{if
}q=\infty.\end{array}\right.$ (21)

(b) The Jackson-type inequality

$f^{*}(t)\leq
t^{1-1/\theta}C_{\theta,q}\|f\|_{q\theta^{2}/(1-\theta)}\quad\text{for
all}\quad{f\in L_{\mu}^{q\theta^{2}/(1-\theta)}},$ (22)

as well as, the following bilateral Bernstein-Jackson-type inequalities are
valid,

$\displaystyle t^{-1+1/\theta}f^{*}(t)$
$\displaystyle\leq{C}_{\theta,q}\|f\|_{q\vartheta^{2}/(1-\theta)}\leq
2^{1/2\theta}\|f\|^{-1+1/\theta}_{0}\|f\|_{\infty}$ (23)

for all $f\in L^{0}_{\mu}\cap L^{\infty}_{\mu}$.

Proof: (a) By definition, the functional $E(t,f)$ is the infimum of
$\|f-f_{0}\|_{\infty}$ such that
${\mu(\operatorname{\operatorname{supp}}f_{0})\leq t}$. The next reasoning
extends (bergh76, , Lemma 7.2.1) on the case of $\mathfrak{A}$-valued
functions.

Put $g_{0}(x)=f(x)$ on the subset $\operatorname{\operatorname{supp}}f$ and
let $g_{0}(x)=0$ outside $\operatorname{\operatorname{supp}}f$. Then we obtain
${\|f-g_{0}\|_{\infty}}\leq{\|f-g\|_{\infty}}$.

In a similar way, let $g_{\sigma}(x)=f(x)$ if $|f(x)|>\sigma$ and
${g_{\sigma}(x)=0}$ otherwise. Then for the number
$\tau={\sup\big{\\{}|f(x)|\colon
x\neq\operatorname{\operatorname{supp}}f\big{\\}}}$ we get
$\operatorname{\operatorname{supp}}f_{\tau}\subset\operatorname{\operatorname{supp}}f$.
It follows ${\mu(\operatorname{\operatorname{supp}}g_{\tau})\leq t}$. Since
$\|f-g_{\tau}\|_{\infty}\leq\tau$ and $\|f-g_{0}\|_{\infty}=\tau$, we obtain

$E\big{(}t,f\mathchar
24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)}={\inf}_{\sigma}\left\\{\|f-g_{\sigma}\|_{\infty}\colon\mu\left(\operatorname{\operatorname{supp}}g_{\sigma}\right)\leq
t\right\\},$

where the rigth hand side is equal to $f^{*}$. As a result, the equality (20)
holds.

Using the best approximation $E$-functional, we define the quasi-normed space

$\displaystyle E_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$
$\displaystyle=\left\\{f\in{L^{0}_{\mu}+L^{\infty}_{\mu}}\colon\|f\|_{E_{\theta,q}}<\infty\right\\},$
(24) $\displaystyle\|f\|_{E_{\theta,q}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{-1+1/\theta}E(t,f)\right]^{q\theta}\frac{dt}{t}\right)^{1/q\theta}&\hbox{if
}q<\infty,\\\\[8.61108pt]
\sup\limits_{0<t<\infty}t^{-1+1/\theta}E(t,f)&\hbox{if
}q=\infty.\end{array}\right.$ (27)

By the known approximation theorem (see (bergh76, , Thm 7.2.2)) the isometric
isomorphism

$E_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=L_{\mu}^{q\theta^{2}/(1-\theta)},\quad\|f\|_{E_{\theta,q}}=\|f\|_{\vartheta^{2}q/(1-\theta)}$
(28)

holds for all ${f\in L_{\mu}^{q\theta^{2}/(1-\theta)}}.$ Combining equalities
(20), (24) and (28), we get the equality (21).

(b) In what follows, we use the classic integral of the real interpolation
method

$\|f\|_{\theta,q}=\left(\int_{0}^{\infty}\left|t^{-\theta}f(x)\right|^{q}\frac{dt}{t}\right)^{1/q},\quad{0<\theta<1},\
0<q<\infty.$ (29)

Let us consider the quadratic $K_{2}$-functional (see e.g. (McLean2000, , App.
B)) for the interpolation couple of quasi-normed groups
$\left(L^{0},L^{\infty}\right)$,

$\displaystyle K_{2}(t,f)$ $\displaystyle={K}_{2}\big{(}t,f\mathchar
24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$
$\displaystyle=\inf_{f=f_{0}+f_{\infty}}\left\\{\left(\|f_{0}\|_{0}^{2}+t^{2}\|f_{\infty}\|_{\infty}^{2}\right)^{1/2}\colon
f_{0}\in L^{0}_{\mu},f_{\infty}\in L^{\infty}_{\mu}\right\\},$

determining the real interpolation quasi-normed Abelian group

$\displaystyle\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}_{\theta,q}:=K_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=\left\\{f=f_{0}+f_{\infty}\colon\|K_{2}(\cdot,f)\|_{\theta,q}<\infty\right\\}$

which is endowed with the norm

$\displaystyle\|f\|_{K_{\theta,q}}$
$\displaystyle=\left\\{\begin{array}[]{ll}N_{\theta,q}\|K_{2}(\cdot,f)\|_{\theta,q}&\hbox{if
}q<\infty\\\\[6.45831pt]
\sup\limits_{t\in(0,\infty)}t^{-\theta}K_{2}(t,f)&\hbox{if
}q=\infty.\end{array}\right.$

Note that for any ${0<\vartheta<1}$ and ${q=2}$ the following equalities hold,

$\displaystyle
N_{\theta,2}=\|\mathcal{N}\|_{\theta,2}^{-1}=\|K_{2}(\cdot,1)\|_{\theta,2}^{-1}=\left(\frac{2\sin\pi\theta}{\pi}\right)^{1/2}.$

In fact, the relationship between the weight function
$\mathcal{N}(t)^{2}=t^{2}/(1+t^{2})$ and the quadratic $K_{2}$-functional is
explained by the formula

$N_{\theta,2}=\|\mathcal{N}\|_{\theta,2}^{-1}=\|K_{2}(\cdot,1)\|_{\theta,2}^{-1}$

(see e.g. (McLean2000, , Ex. B.4)). It follows from

$\min_{z=z_{0}+z_{1}}\left(\alpha_{0}|z_{0}|^{2}+\alpha_{1}|z_{1}|^{2}\right)={\alpha_{0}\alpha_{1}|z|^{2}}{\alpha_{0}+\alpha_{1}}$

for a fixed $\alpha_{0},\alpha_{1}>0$ and a complex $z$. This minimum is
achieved when

$\alpha_{0}z_{0}=\alpha_{1}z_{1}=\frac{\alpha_{0}\alpha_{1}z}{(\alpha_{0}+\alpha_{1})}.$

Thus, $K_{2}(t,1)$ is minimized when $f_{0}$, $f_{1}$ are such that

$f_{0}=t^{2}f_{1}=\frac{t^{2}}{1+t^{2}}.$

By integrating the above functions (see e.g. (McLean2000, , Ex. B.5, Thm B.7))
it follows that the normalization factor $N_{\theta,2}$ is equal to
$N_{\theta,2}=\left({2\sin\pi\theta}/{\pi}\right)^{1/2}.$

In particular, the equalities (20),

$\displaystyle f^{*}(t)$ $\displaystyle=\inf\big{\\{}\sigma\colon
m(\sigma,f_{\sigma})\leq t\big{\\}}$
$\displaystyle=\inf\left\\{\|f-f_{0}\|_{\infty}\colon\|f_{0}\|_{0}<t\right\\}=E\big{(}t,f\mathchar
24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)},\quad
f\in{L_{\mu}^{\infty}},$

as well as, the isometric isomorphism (28),

$\displaystyle
E_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=L_{\mu}^{q\theta^{2}/(1-\theta)},\quad\|f\|_{E_{\theta,q}}=\|f\|_{\vartheta^{2}q/(1-\theta)}$

allows calculating the exact form of best constants (19).

######  Corollary 8.

The decreasing rearrangement for $q=2$ has the estimation

$\displaystyle f^{*}(t)$ $\displaystyle\leq
t^{1-1/\theta}\left(\frac{\sin\pi\theta}{\pi\theta}\right)^{1/2\theta}\\!\|f\|_{2\theta^{2}/(1-\theta)}\quad\text{for
all}\quad{f\in L_{\mu}^{2\theta^{2}/(1-\theta)}}$ (30)

which for $\theta=1/2$ can be written as follows

$\displaystyle f^{*}(t)$ $\displaystyle\leq\frac{2}{\pi
t}\,\int|f|\,\mu(dx)\quad\text{for all}\quad f\in L^{1}_{\mu}.$ (31)

Note that if $\theta\to 1$ then $\sin\pi\theta\to 0$ thus above inequalities
disappear. One has a sense only for the quasi-normed group
$L_{\mu}^{2\theta^{2}/(1-\theta)}$ with $0<\theta<1$.

######  Corollary 9.

The equality (21) with $s+1=1/\vartheta$ and $\tau=\vartheta q$ takes the form

$\|f\|_{\tau/s}=\left\\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{s}f^{*}(t)\right]^{\tau}\frac{dt}{t}\right)^{1/\tau}&\hbox{if
}\tau<\infty\\\\[8.61108pt] \sup\limits_{0<t<\infty}t^{s}f^{*}(t)&\hbox{if
}\tau=\infty\end{array}\right.$

for any $f\in L^{\tau/s}_{\mu}$. Then the Jackson-type inequality takes the
form

$f^{*}(t)\leq t^{-s}2^{(s+1)/2}\,c_{s,\tau}\|f\|_{\tau/s}\quad\text{for
all}\quad{f\in L^{\tau/s}_{\mu}}$ (32)

and bilateral Bernstein-Jackson-type inequalities take the form

$\displaystyle t^{s}2^{-(s+1)/2}f^{*}(t)$ $\displaystyle\leq
c_{s,\tau}\|f\|_{\tau/s}\leq\|f\|^{s}_{0}\|f\|_{\infty}\quad\text{for
all}\quad{f\in L^{0}_{\mu}\cap L^{\infty}_{\mu}}.$ (33)

It instantly follows from the relations

$C_{\theta,q}=\left\\{\begin{array}[]{cl}2^{(s+1)/2}\,c_{s,\tau}&\hbox{if
}\tau<\infty\\\ 2^{(s+1)/2}&\hbox{if
}\tau=\infty\end{array}\right.,\quad\text{where}\quad\vartheta=\frac{1}{s+1},\quad
2^{1/2\theta}=2^{(s+1)/2}.$

######  Remark 2.

The inequalities (33) is an extension on the case measurable functions the
known best approximation Bernstein-Jackson inequalities. Whereas the
inequality (32) is an extension of the approximation Jackson inequality. The
scale of approximation quasi-normed Abelian groups is often denoted as

$\mathcal{B}_{\tau}^{s}(L^{0}_{\mu},L^{\infty}_{\mu}):=E_{\theta,q}(L^{0}_{\mu},L^{\infty}_{\mu}),$

where the space $\mathcal{B}_{\tau}^{s}$ coincides with a suitable extension
of the Besov type quasi-normed groups (see e.g. (bergh76, , Thm 7.2.4),
(Triebel78, , Def. 4.2.1/1)). The scale of quasi-normed Besov Abelian groups
for another approximation couples is described in DL19 .

##  4 Examples of bilateral estimates with exact constants

Taking into account the previous statements, we can give typical examples of
Bernstein-Jackson inequalities with explicit constants for different types of
best approximations.

######  Example 1 (Applications to classic Besov scales).

Let ${\mathfrak{A}_{0}=L^{p}(\mathbb{R}^{n})}$ with ${(1<p<\infty)}$. As is
known (see e.g. Nikolski75 ) each real-valued function ${a\in
L^{p}(\mathbb{R}^{n})}$ can be approximated by entire analytic functions
$g\in\mathcal{E}_{p}^{t}(\mathbb{C}^{n})$, where
$\mathcal{E}_{p}^{t}(\mathbb{C}^{n})$ means the space of entire analytic
functions on $\mathbb{C}^{n}$ of an exponential type ${t>0}$ with restrictions
to $\mathbb{R}^{n}$ belonging to $L^{p}(\mathbb{R}^{n})$. The best
approximations can be characterized by the best approximation functional

$E\left(t,a\mathchar
24635\relax\;\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)=\inf_{t>0}\left\\{\|a-g\|_{L^{p}(\mathbb{R}^{n})}\colon
g\in\mathcal{E}_{p}^{t}(\mathbb{C}^{n})\right\\},$

where the subspace ${\mathfrak{A}_{1}=\mathcal{E}_{p}}$ with
$\mathcal{E}_{p}={\bigcup}_{t>0}\mathcal{E}_{p}^{t}(\mathbb{C}^{n})$ is
endowed with the quasinorm

$|g|_{\mathcal{E}_{p}}=\|g\|_{L^{p}(\mathbb{R}^{n})}+\left\\{\sup{|\zeta|\colon\zeta\in\operatorname{\operatorname{supp}}\hat{g}}\right\\},\quad
g\in\mathcal{E}_{p}$

defined using the support $\operatorname{\operatorname{supp}}\hat{g}$ of the
Fourier-image $\hat{g}$. Then according to Theorem 1 the corresponding
approximation inequalities take the form

$\displaystyle t^{s}E\left(t,a\mathchar
24635\relax\;\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)$ $\displaystyle\leq
c_{s,\tau}\|f\|_{B_{p,\tau}^{s}(\mathbb{R}^{n})}\leq
2^{1/2}|a|^{s}_{\mathcal{E}_{p}}\|a\|_{L^{p}},\ {a\in\mathcal{E}_{p}\cap
L^{p}(\mathbb{R}^{n})},$ $\displaystyle E\left(t,a\mathchar
24635\relax\;\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)$
$\displaystyle\leq{t^{-s}c_{s,\tau}}\,\|a\|_{B_{p,\tau}^{s}(\mathbb{R}^{n})},\quad{a\in{B_{p,\tau}^{s}(\mathbb{R}^{n})}},$

In this case the approximation scale

$\mathcal{B}_{\tau}^{s}\big{(}\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\big{)}=\left(\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)_{\vartheta,q}^{1/\vartheta}$

exactly coincides with the classic Besov scale denoted by
$B_{p,\tau}^{s}(\mathbb{R}^{n})$ (see (Triebel78, , p.197)).

######  Example 2 (Applications to scales of periodic functions).

Following e.g. (bergh76, , no 1.5), Prestin  we can write Bernstein-Jackson
inequalities in a more general form. Let $X=\mathbb{T}$ be the $1$-dimensional
torus and the Hilbert space $\mathfrak{A}_{1}=L^{2}(\mathbb{T})$ has the
orthonormal basis $\left\\{\mathfrak{e}_{k}=e^{2\pi\mathrm{i}kt}\colon
k\in\mathbb{Z},\,t\in\mathbb{T}\right\\}$. Let
$\mathfrak{A}_{0}=\\{\text{trigonometric polynomials \ }a_{0}\\}$ with the
quasi-norm $|a_{0}|_{0}=\operatorname{\operatorname{deg}}a_{0}$, and
$\mathfrak{A}_{1}=\\{\text{$2\pi$-periodic functions \ }a\\}$ with the norm
$|a|_{1}=\|a\|_{L^{2}(\mathbb{T})}$, as well as, $E(n,a\mathchar
24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\left\\{\|a-a_{0}\|_{L^{2}(\mathbb{T})}\colon\operatorname{\operatorname{deg}}a_{0}<n\right\\}$
for all ${a\in L^{2}(\mathbb{T})}$. Let
$c_{j,\tau}:=\left\\{\begin{array}[]{cl}\displaystyle\left[\frac{j}{\tau(j+1)^{2}}\right]^{1/\tau}&\hbox{if
}\tau<\infty\\\ \displaystyle 1&\hbox{if }\tau=\infty\end{array}\right.$ with
$j\in\mathbb{N}$. Then Bernstein-Jackson inequalities have the form

$\displaystyle n^{j}E(n,a)$
$\displaystyle\leq{c}_{j,\tau}|a|_{\mathcal{B}_{\tau}^{j}}\leq
2^{1/2}|a|^{j}_{0}|a|_{1}\quad\text{for all}\quad
a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1},$ $\displaystyle E(n,a)$
$\displaystyle\leq
n^{-j}{c}_{j,\tau}|a|_{\mathcal{B}_{\tau}^{j}}\quad\text{for
all}\quad{a\in\mathcal{B}_{\tau}^{j}(\mathfrak{A}_{0},\mathfrak{A}_{1})},$

where
$\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta}$
with $j+1=1/\vartheta\in\mathbb{N}$, $\tau=\vartheta q$.

Let $\mathfrak{A}=L^{2}(\mathbb{T})$. Consider the Banach space $L^{p}_{\mu}$
of $L^{2}(\mathbb{T})$-valued functions ${f\colon X\to L^{2}(\mathbb{T})}$.
Then bilateral Bernstein-Jackson inequalities have the form

$\displaystyle n^{-j}2^{-(j+1)/2}f^{*}(n)$ $\displaystyle\leq
c_{j,\tau}\|f\|_{\tau/j}\leq\|f\|^{j}_{0}\|f\|_{\infty},\quad{f\in
L_{\mu}^{0}\cap L_{\mu}^{\infty}},$ $\displaystyle f^{*}(n)$
$\displaystyle\leq n^{j}2^{(j+1)/2}c_{j,\tau}\|f\|_{\tau/j},\quad{f\in
L_{\mu}^{\tau/j}},$

where $f^{*}(n)=E\big{(}n,f\mathchar
24635\relax\;L_{\mu}^{0},L_{\mu}^{\infty}\big{)}=\inf\big{\\{}\sigma\colon
m(\sigma,f_{\sigma})\leq t\big{\\}}$ for all $f\in{L_{\mu}^{\infty}}$.

######  Example 3 (Applications to operator spectral approximations).

Consider the example commonly used in foundations of quantum systems for
describing quantum states (see e.g. (Hall2013, , no 3.4)). Similar estimates
of spectral approximations were early analyzed in the paper DL19 . In what
follows, we consider the spaces $L^{p}=L^{p}(\mathbb{R}\mathchar
24635\relax\;\mathbb{R})$ of real-valued functions.

Let $H$ be a Hilbert complex space with the norm
$\|\cdot\|_{H}={\langle\cdot\mid\cdot\rangle^{1/2}}$ and a self-adjoint
unbounded linear operator $T$ with the dense domain $\mathcal{D}(T)\subset H$
is given. By spectral theorem the measurable function ${f}$ from $T$ can be
well defined using the following spectral expansions (see e.g. Hall2013 )

$T=\int_{\sigma(T)}\lambda\,\mu(d\lambda),\quad{f}(T)=\int_{\sigma(T)}{f}(\lambda)\,\mu(d\lambda),$

where $\mu$ is a unique projection-valued measure determined on its spectrum
$\sigma(T)\subset\mathbb{R}$ with values in the Banach space of bounded linear
operators $\mathcal{L}(H)$, which can be extended on
$\mathbb{R}\setminus\sigma(A)$ as zero.

At beginning, we consider the case when for any Borel set
$\varSigma\subset\mathbb{R}$ and any $\psi\in\mathcal{D}(A)$ the
$\mathcal{L}(H)$-valued measurable function

$\varSigma\longmapsto
f(\varSigma)=\int_{\varSigma}{f}(\lambda)\,\mu_{\psi}(d\lambda)$

belongs to $L^{\infty}$ with respect to the positive measure
$\mu_{\psi}(d\lambda):=\big{\langle}\mu(d\lambda)\psi\mid\psi\big{\rangle}$,
where $\|\psi\|_{H}=1$.

Consider the corresponding quadratic forms
$f_{\psi}(\varSigma)=\big{\langle}f(\varSigma)\psi\mid\psi\big{\rangle}$ as
measurable functions of $L^{2}$ and let $f^{*}_{\psi}$ be the suitable
decreasing rearrangement. By Theorem 14 the inequalities (32) and (33) in this
case take the form

$\displaystyle f_{\psi}^{*}(t)$ $\displaystyle\leq
t^{1-1/\theta}C_{\theta,q}\|f_{\psi}\|_{q\theta^{2}/(1-\theta)},\quad{f_{\psi}\in
L_{\mu}^{q\theta^{2}/(1-\theta)}},$ (34) $\displaystyle
t^{-1+1/\theta}f_{\psi}^{*}(t)$
$\displaystyle\leq{C}_{\theta,q}\|f_{\psi}\|_{q\vartheta^{2}/(1-\theta)}\leq
2^{1/2\theta}\|f_{\psi}\|^{-1+1/\theta}_{0}\|f_{\psi}\|_{\infty},\quad{f_{\psi}\in
L^{0}_{\mu}\cap L^{\infty}_{\mu}},$ (35)

where
$\|f_{\psi}\|_{\infty}=\sup_{\varSigma\subset\mathbb{R}}|f_{\psi}(\varSigma)|$
and
$\|f_{\psi}\|_{0}=\mu_{\psi}\left(\operatorname{\operatorname{supp}}f_{\psi}\right)$.
Applying the estimation (31) from Corollary 8, we obtain

$\displaystyle f_{\psi}^{*}(t)$ $\displaystyle\leq\frac{2}{\pi
t}\,\int|f(\lambda)|\,\mu(d\lambda)\quad\text{for all}\quad f_{\psi}\in
L^{1}_{\mu}.$ (36)

The inequality (34), (36) give error estimations (in quadratic forms terms) of
spectral approximations $f_{\psi}(T)={\langle f(T)\psi\mid\psi\rangle}$ by
$f_{\psi}(\varSigma)$ with $\mu\left(\varSigma\right)\leq t$. The inequalities
(35) characterise the approximation accuracy.

In another case, using the $\mathcal{L}(H)$-valued measurable uniformly
bounded functions

$\tilde{f}\colon\varSigma\longmapsto\int_{\varSigma}f(\lambda)\,\mu(d\lambda),\quad
f\in L^{1}_{\mu}$

belonging to $L^{1}_{\mu}(\mathbb{R},\mathcal{L}(H))$, we same as above obtain
the inequality

$\displaystyle\tilde{f}^{*}(t)$ $\displaystyle\leq\frac{2}{\pi t}\Big{\|}\int
f(\lambda)\,\mu(d\lambda)\Big{\|}_{\mathcal{L}(H)}.$

######  Example 4 (Samples of numerical calculations).

The inequality (32) together with the exact value of the normalized constant
${c}_{s,\tau}$ are suitable for direct numerical calculations. Illustrate this
on several graphs, taking as an example the inverse Gaussian distribution
function.

Consider two-parameter inverse Gaussian distribution with support on
$(0,\infty)$ that is given by

$f(t)=C\sqrt{\frac{l}{2\pi
t^{3}}}\exp\left(-\frac{l(t-m)^{2}}{2m^{2}t}\right),\quad C,t>0$

with the mean $m>0$ and the shape parameter $l>0$. By Theorem 14

$\displaystyle f^{*}(u)=E(u,f)$
$\displaystyle=\inf\left\\{\|f-f_{\sigma}\|_{\infty}\colon
m(\sigma,f_{\sigma})\leq u\right\\},$ $\displaystyle m(\sigma,f_{\sigma})$
$\displaystyle=\mu\left\\{u\in(0,\infty)\colon{|f_{\sigma}(u)|>\sigma}\right\\}$

coincides with the non-increasing rearrangement of $f$ on $(0,\infty)$. As a
result, we obtain

$\displaystyle f^{*}(u)=E(u,f)$
$\displaystyle\leq{c}_{s,\tau}u^{-s}\,\|f\|_{\tau/s}\quad\text{for
all}\quad{f\in L_{\mu}^{\tau/s}},$

where
$L_{\mu}^{\tau/s}=\mathcal{B}_{\tau}^{s}\left(L^{0}_{\mu},L^{\infty}_{\mu}\right)$
in accordance with Remark 2 and the equality (21).

The obtained results of numerical calculations are illustrated on Fig. 2,3,4.

Figure 2: Graphs of the function $f(t)=10\sqrt{\frac{2}{\pi
t^{3}}}e^{-\frac{(t-2)^{2}}{2t}}$ (dots) and its non-increasing rearrangement
$f^{*}(t)$ (polygon), (${0<t<10}$) Figure 3: Graphs of approximation $E(u,f)$
via $u^{-s}c_{s,\tau}\|f\|_{\mathcal{B}_{\tau}^{s}}$ (${0<u<11}$,
${s=\tau=2}$) using Jackson’s inequality

## Statements and Declarations: There are no conflicts and potential competing
of interest to disclose.

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