# Bernstein–Nikolskii inequality:
optimality with respect to the smoothness parameter

Michael I. Ganzburg  M. I. Ganzburg, 212 Woodburn Drive, Hampton, VA 23664
<EMAIL_ADDRESS>,  Miquel Saucedo  M. Saucedo, Centre de Recerca
Matemàtica
Campus de Bellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain;
<EMAIL_ADDRESS>and  Sergey Tikhonov  S. Tikhonov, ICREA, Pg. Lluís
Companys 23, 08010 Barcelona, Spain
Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C 08193 Bellaterra
(Barcelona), Spain, and Universitat Autònoma de Barcelona, Edifici C 08193
Bellaterra (Barcelona), Spain<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the form of the constant $C$ in the
Bernstein–Nikolskii inequalities $\|f^{(s)}\|_{q}\lesssim
C(s,p,q)\left\|f\right\|_{p},\,0<p<q\leqslant\infty$, for trigonometric
polynomials and entire functions of exponential type. We obtain the optimal
behavior of the constant with respect to the smoothness parameter $s$.

###### Key words and phrases:

Berstein–Nikolskii inequality, trigonometric polynomial, entire function of
exponential type, discrete Hardy space, $H_{p}$-atom

###### 2010 Mathematics Subject Classification:

Primary 41A17, Secondary 42A05

M. Saucedo is supported by the Spanish Ministry of Universities through the
FPU contract FPU21/04230. S. Tikhonov is supported by PID2020-114948GB-I00,
2021 SGR 00087, AP 14870758. This work is supported by the CERCA Programme of
the Generalitat de Catalunya, and the Severo Ochoa and María de Maeztu Program
for Centers and Units of Excellence in R&D (CEX2020-001084-M)

##  1\. Introduction

###  1.1. Notation and Preliminaries

Let us first define the $L_{p}$\- and $\ell_{p}$-quasinorms

$\displaystyle\begin{array}[]{lll}\|f\|_{p}:=\left(\int_{\mathbb{R}}|f(x)|^{p}dx\right)^{1/p},&\|f\|_{p}^{*}:=\left(\int_{-\pi}^{\pi}|f(x)|^{p}dx\right)^{1/p},&p\in(0,{\infty}];\\\
\|a\|_{\ell_{p}(\mathbb{Z})}:=\left(\sum_{k\in\mathbb{Z}}\left|a_{k}\right|^{p}\right)^{1/p},&\|{\lambda}\|_{\ell_{p}(\mathbb{Z}_{+})}:=\left(\sum_{d=0}^{\infty}\left|{\lambda}_{d}\right|^{p}\right)^{1/p},&p\in(0,{\infty}].\end{array}$

The Fourier transform of an integrable function $\phi$ and its inverse Fourier
transform are defined by the standard formulae

$\displaystyle\widehat{\phi}(\xi):=\int_{\mathbb{R}}\phi(x)e^{-ix\xi}dx,\qquad\check{\phi}(\xi):=\frac{1}{2\pi}\int_{\mathbb{R}}\phi(x)e^{ix\xi}dx.$

Next, let $\mathcal{T}_{n}$ be the space of all trigonometric polynomials
$T_{n}(x)=\sum_{k=-n}^{n}c_{k}e^{ikx}$ of degree at most $n$ with complex
coefficients. We also use the Dirichlet kernel
$D_{N}(x):=1+2\sum_{k=1}^{N}\cos kx$ of order $N\in\mathbb{N}$ on several
occasions.

In addition, let $E_{{\sigma},p}$ be the Paley–Wiener space of all complex-
valued entire functions $f$ of exponential type ${\sigma}>0$ with
$\|f\|_{p}<{\infty}$ (recall that an entire complex-valued function $f$ is of
exponential type ${\sigma}$ if for any $\varepsilon>0$ there exists a constant
$C_{\varepsilon,f}$ such that $|f(z)|\leqslant
C_{\varepsilon,f}e^{({\sigma}+\varepsilon)|z|}$ for all complex $z$). It is
well known that $E_{{\sigma},p}\subset E_{{\sigma},q}$, if
$0<p<q\leqslant{\infty}$, and $E_{{\sigma},p}$ is a Banach space for
$p\in[1,{\infty}]$ and a quasi-Banach space for $p\in(0,1)$. Throughout the
paper, we use the notation $E_{p}:=E_{1,p}$.

In the paper, we often use without referencing the following fundamental
result by Paley and Wiener: $f\in E_{2}$ if and only if there exists $\psi$
with $\left\lVert\psi\right\rVert_{2}<{\infty}$ such that $\psi$ is supported
in $[-1,1]$ and $f(\xi)=\widehat{\psi}(\xi)$.

The sinc function $\operatorname{sinc}x:=\sin x/x$, the floor function
$\lfloor x\rfloor$, the ceiling function $\lceil x\rceil$, the gamma function
$\Gamma(z)$, and the beta function

(1.2)
$B(z_{1},z_{2}):=\int_{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}dt=\frac{\Gamma(z_{1})\Gamma(z_{2})}{\Gamma(z_{1}+z_{2})},\qquad
z_{1}>0,\quad z_{2}>0,$

are used as well. The following asymptotic formula for the quotient of the
gamma functions is well-known (see [10, Sect. 1.18, Eqn. (5)]):

(1.3) $\lim_{z\to{\infty}}z^{b-a}\frac{\Gamma(z+a)}{\Gamma(z+b)}=1,\qquad
z>0,\quad a,b\in\mathbb{R}.$

We also use the notations $\lesssim,\,\gtrsim$, and $\asymp$ in the following
sense: ${\varphi}(\tau,\gamma,\ldots)\lesssim\delta(\tau,\gamma,\ldots)$,
${\varphi}(\tau,\gamma,\ldots)\gtrsim\delta(\tau,\gamma,\ldots),$ and
${\varphi}(\tau,\gamma,\ldots)\asymp\delta(\tau,\gamma,\ldots)$ mean that
there exist constants $C_{1}>0$ and/or $C_{2}>0$ independent of the essential
parameters $\tau,\,\gamma,\ldots$ such that
${\varphi}(\tau,\gamma,\ldots)\leqslant
C_{1}\delta(\tau,\gamma,\ldots),\,{\varphi}(\tau,\gamma,\ldots)\geqslant
C_{2}\delta(\tau,\gamma,\ldots)$, and
$C_{2}\delta(\tau,\gamma,\ldots)\leqslant{\varphi}(\tau,\gamma,\ldots)\leqslant
C_{1}\delta(\tau,\gamma,\ldots),$ respectively, for the relevant ranges of
$\tau,\gamma,\ldots$. The dependence of the constants on certain parameters is
indicated by using subscripts, e.g., $\lesssim_{p,q}$ or $\asymp_{p,q}$. The
absence of the subscripts means that $C_{1}$ and/or $C_{2}$ are absolute
constants.

###  1.2. Bernstein–Nikolskii Inequalities

The following Bernstein–Nikolskii inequalities:

(1.4)
$\displaystyle\left\|T_{n}^{(s)}\right\|^{*}_{q}\lesssim_{s,p,q}n^{s+1/p-1/q}\left\|T_{n}\right\|^{*}_{p},\qquad
T_{n}\in\mathcal{T}_{n};$ (1.5)
$\displaystyle\left\|f^{(s)}\right\|_{q}\lesssim_{s,p,q}{\sigma}^{s+1/p-1/q}\left\|f\right\|_{p},\qquad
f\in E_{{\sigma},p};$

for $0<p\leqslant q\leqslant{\infty}$ and $s=0,\,1,\ldots$, have been known
for more than sixty years.

Both (1.4) and (1.5) are fundamental results in pure and applied harmonic
analysis. The applications include various topics in approximation theory,
function analysis, and PDE’s.

Various estimates and asymptotics for the constants in (1.4) and (1.5) were
discussed in numerous publications (see, e.g., [12, 18, 14, 11, 6, 3, 4] and
references therein). In particular, the sharp constants in (1.4) and (1.5) are
known for $q={\infty}$ and $p=2$ (see [11, Eqns. (1.5) and (1.13)]). The first
and third authors [11] discussed limit relations between the sharp constants
in (1.4) and (1.5). Bang and Huy [3, Theorem 2.1] studied the influence of a
large $s$ on the constant in (1.5). Simonov and Glazyrina [18, Theorem 1]
studied the sharp constant in the closely related to (1.4) Markov-Nikolskii
inequality for $q={\infty}$ and $p=1$. In the case $s=0$ and $p<q=\infty$,
estimate (1.5) was recently investigated in [14, 6, 4].

Importantly, the problem of finding the asymptotic behavior of the constant in
(1.4) as $n\to{\infty}$ remains open.

In this paper, we obtain new estimates of the constants in (1.4) and (1.5)
that cannot be improved on the spaces $\mathcal{T}_{n}$ and $E_{{\sigma},p}$
with respect to the parameter $s\in\mathbb{N}$.

###  1.3. Main Results

######  Theorem 1.1.

For every $n,s\in\mathbb{N}$ and $0<p<q\leqslant\infty$, the following
relation holds:

(1.6) $\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\ \left\lVert
T_{n}\right\rVert_{p}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}\asymp_{p,q}n^{s}\left(1+\left({n}/{s}\right)^{{1}/{p}-{1}/{q}}\right).$

######  Theorem 1.2.

For every $s\in\mathbb{N}$ and $0<p<q\leqslant\infty$, the following relation
holds:

(1.7) $\sup_{\begin{subarray}{c}f\in E_{{\sigma},p},\\\ \left\lVert
f\right\rVert_{p}=1\end{subarray}}\left\lVert
f^{(s)}\right\rVert_{q}\asymp_{p,q}{\sigma}^{s}({\sigma}/s)^{{1}/{p}-{1}/{q}}.$

For a certain subset of $\mathcal{T}_{n}$, relation (1.6) for $q={\infty}$ and
$p=1$ can be improved.

######  Definition 1.3.

We say that a finite sequence of numbers $c=(c_{0},\dots,c_{n})$ is concave if
$c_{0}\geqslant c_{1}\geqslant\dots\geqslant c_{n}\geqslant c_{n+1}:=0$ and
$c_{n}-c_{n+1}=:\Delta c_{n}\geqslant\Delta
c_{n-1}\geqslant\dots\geqslant\Delta c_{0}=c_{0}-c_{1}$.

The next theorem shows that by restricting ourselves just to concave sequences
of coefficients, we come to the better behavior in $s$ and $n$ on the right-
hand side of (1.6). Let $\mathcal{T}_{n,\mbox{c}}\subset\mathcal{T}_{n}$ be
the set of all trigonometric polynomials
$T_{n}(x)=c_{0}+\sum_{k=1}^{n}2c_{k}\cos{kx}$ with a concave sequence $c$.

######  Theorem 1.4.

For every $n,s\in\mathbb{N}$, the following relation holds:

(1.8) $\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n,\mbox{c}},\\\
\left\lVert T_{n}\right\rVert_{1}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{\infty}^{*}\asymp\left(n^{s}+\frac{n^{s+1}}{s+1}\right)\left(\frac{1}{\log{(s+2)}}+\frac{1}{\log(n+2)}\right).$

######  Remark 1.5.

A weighted version of the $\lesssim$ part of Theorem 1.2 for $1\leqslant
p<q\leqslant{\infty}$ with power weights was established by Bang and Huy in
[3, Theorem 2.1].

######  Remark 1.6.

Analyzing the proofs, we note that all theorems above are valid for the
fractional case, namely, when $s\in(0,{\infty})$. Here, the derivatives of a
trigonometric polynomial are defined in the Weyl sense, i.e.,
$T^{(s)}(x):=\sum_{|k|\leqslant n}(ik)^{s}c_{k}e^{ikx}$, where
$(ik)^{s}=|k|^{s}e^{i\frac{\pi}{2}s\mathop{\rm sgn}\nolimits k}$. The similar
definition for functions from $f\in E_{\sigma,p}$ can be found in [17].

The proofs of the lower and upper estimates in (1.6) and (1.7) are presented
in Sections 2 and 3, respectively. The properties of concave sequences and
trigonometric polynomials from $\mathcal{T}_{n,\mbox{c}}$, which are needed
for the proof of Theorem 1.4, are discussed in Section 4. Section 5 contains
the proofs of all theorems. In Section 6 we discuss several properties of
extremal trigonometric polynomials in the Bernstein–Nikolskii inequality for
$p=1$ and $q=\infty$. We note that it suffices to prove Theorem 1.2 for
${\sigma}=1$, so we discuss functions from $E_{p},\,p\in(0,{\infty}]$, in
Sections 2, 3, 5, and 7.

For $p\in[1,{\infty}]$, the $\lesssim$ part of Theorem 1.1 is proved by the
standard method of estimation of $\left\lVert
D_{n}^{(s)}\right\rVert_{r}^{*}$. The proof of the $\lesssim$ part of Theorem
1.1 for $p\in(0,1)$ is more complicated. It is based on an inequality for
functions from $E_{p}$ that is obtained by using certain properties of the
discrete Hardy spaces, which are discussed in the Appendix of Section 7. The
proof of the $\lesssim$ part of Theorem 1.2 for $p\in(0,{\infty}]$ uses the
$\lesssim$ part of Theorem 1.1 and also Theorem 1.4 from [11]. The lower
estimates in Theorems 1.1 and 1.2 are proved by the construction of special
functions from $E_{p}$ and $\mathcal{T}_{n}$ in the spirit of [1, Sect. 84].

Finally, it is worth mentioning that the following improvements of the
$\lesssim$ parts of relations (1.6) and (1.7) are valid in the case
$p\in(0,1]$ and $q=\infty$: for any $T_{n}(x)=\sum_{k=-n}^{n}c_{k}e^{ikx}$ and
any $f\in E_{p}$,

$\displaystyle\sum_{k=-n}^{n}|k|^{s}|c_{k}|\lesssim_{p}n^{s}\left(1+\left({n}/{s}\right)^{1/p}\right)\left\lVert
T_{n}\right\rVert_{p}^{*},\qquad\int_{-1}^{1}|\xi|^{s}|\widehat{f}(\xi)|\,d\xi\lesssim_{p}s^{-1/p}\left\lVert
f\right\rVert_{p}$

(see Lemmas 3.5 and 3.3, respectively).

##  2\. Lower Bounds in Theorems 1.1 and 1.2

In this section, we prove the $\gtrsim$ parts of Theorems 1.1 and 1.2.

######  Lemma 2.1.

For every $s\in\mathbb{N}$ and $0<p<q\leqslant\infty$, the following relation
holds:

(2.1) $\sup_{\begin{subarray}{c}f\in E_{p},\\\ \left\lVert
f\right\rVert_{p}=1\end{subarray}}\left\lVert
f^{(s)}\right\rVert_{q}\gtrsim_{p,q}{s}^{{1}/{q}-{1}/{p}}.$

###### Proof.

Let $\phi$ be a non-negative bump function, that is, a smooth function
supported in $[-1,1]$ such that $\phi(x)=1$ for $|x|\leqslant{1}/{2}$. Let us
set

$\displaystyle\phi_{s}(x):=s\phi\left(sx-s+1\right),\qquad s\geqslant 2.$

Since $\phi_{s}$ is a smooth function supported in $[1-{2}/{s},1]$, its
Fourier transform $\widehat{\phi}_{s}$ is also smooth and it belongs to
$E_{p}$ for any $p\in(0,{\infty}]$. Moreover,
$\left|\widehat{\phi}_{s}(\xi)\right|=\left|\widehat{\phi}\left(s^{-1}\xi\right)\right|.$
Therefore,

(2.2)
$\left\lVert\widehat{\phi}_{s}\right\rVert_{p}=\left(\int_{\mathbb{R}}\left|\widehat{\phi}\left(s^{-1}\xi\right)\right|^{p}d\xi\right)^{1/p}=\left\lVert\widehat{\phi}\right\rVert_{p}s^{1/p}.$

Furthermore, if $|\xi|\leqslant s$, then

$\displaystyle\left|\left(\widehat{\phi}_{s}\right)^{(s)}(\xi)\right|$
$\displaystyle=$
$\displaystyle\left|\int_{1-2/s}^{1}s\phi\left(sx-s+1\right)e^{-ix\xi}\,x^{s}dx\right|=$
$\displaystyle=$
$\displaystyle\left|\int_{-1/s}^{1/s}s\phi\left(sx\right)e^{-ix\xi}\,\left(x+1-{1}/{s}\right)^{s}dx\right|$
$\displaystyle\geqslant$
$\displaystyle(1-1/s)^{s}\int_{-1/s}^{1/s}s\phi\left(sx\right)\cos x\xi
dx\gtrsim\left|\int_{-1/s}^{1/s}s\phi\left(sx\right)dx\right|\geqslant 1.$

Hence,

(2.3)
$\left\lVert\left(\widehat{\phi}_{s}\right)^{(s)}\right\rVert_{q}\geqslant\left(\int_{0}^{s}\left|\left(\widehat{\phi}_{s}\right)^{(s)}(\xi)\right|^{q}d\xi\right)^{1/q}\gtrsim
s^{1/q}.$

Finally, (2.1) follows from (2.2) and (2.3). ∎

For trigonometric polynomials, we use a variation of the previous argument,
with the Jackson kernel taking the place of the bump function.

######  Lemma 2.2.

For every $n,s\in\mathbb{N}$ and $0<p<q\leqslant\infty$, the following
relation holds:

(2.4) $\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\ \left\lVert
T_{n}\right\rVert_{p}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}\gtrsim_{p,q}n^{s}\left(1+\left({n}/{s}\right)^{{1}/{p}-{1}/{q}}\right).$

###### Proof.

Note first that

(2.5) $\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\ \left\lVert
T_{n}\right\rVert_{p}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}\geqslant\left\lVert\left(e^{in\cdot}\right)^{(s)}\right\rVert_{q}^{*}/\left\lVert
e^{in\cdot}\right\rVert_{p}^{*}\gtrsim_{p,q}n^{s}.$

Next, let $r=r(p)\in\mathbb{N}$ be the smallest number such that $pr>1$. Then
(2.5) shows that if $n\leqslant 4rs$, (2.4) holds.

Furthermore, assuming that $n>4rs$ for $s\geqslant 2$ and setting
$N:=\lfloor{n/rs}\rfloor\geqslant 4$, we consider the Jackson kernel

$\displaystyle J_{r,N}(x):=\left(\frac{\sin{(N+1/2)x}}{\sin{x/2}}\right)^{r}$

with

(2.6) $\left\lVert J_{r,N}\right\rVert_{p}^{*}=\left\lVert
D_{N}\right\rVert_{rp}^{*r}\asymp_{p}N^{r-1/p}.$

Note that (2.6) easily follows from the Hardy–Littlewood equivalence for
monotone Fourier coefficients; namely, for
$f(x)\sim\sum_{k=0}^{\infty}a_{k}\cos kx$ with a non-increasing
$\\{a_{k}\\}_{k=0}^{\infty}$ we have
$\|f\|_{p}^{p}\asymp_{p}\sum_{k=0}^{\infty}a_{k}^{p}(1+k)^{p-2}$ (see [20, Ch.
XII, Lemma 6.6]).

The Fourier coefficients $\hat{J}_{r,N}(\cdot)$ of $J_{r,N}$ are the
convolution of the coefficients of $J_{1,N}$ with themselves $r$ times. Since
$\hat{J}_{1,N}(m)=1$ for $|m|\leqslant N$ and $0$ otherwise, we deduce that
$\hat{J}_{2,N}(m)\asymp N$ for $|m|\leqslant N$. Likewise, we obtain that
$\hat{J}_{3,N}(m)\asymp N^{2}$ for $|m|\leqslant N/2$. By induction on $r$ we
further obtain that the coefficients $\hat{J}_{r,N}(\cdot)$ are non-negative
and there is a $\lambda=\lambda(r)\in(0,1)$ such that for $N$ large enough,

(2.7) $\widehat{J}_{r,N}(m)\asymp_{r}N^{r-1},\qquad|m|\leqslant\lambda N.$

Note that (2.7) for an even $r$ can be also proved by the following asymptotic
relation (see, e.g., [7, Ch. 2, Eqn. (3.38)]):
$1-\pi\widehat{J}_{r,N}(m)/\left\lVert
J_{r,N}\right\rVert_{1}^{*}\asymp_{r}(m/N)^{2}$.

Next, setting

$\displaystyle T(x):=J_{r,N}(x)e^{i(n-rN)x},$

we see that $T\in\mathcal{T}_{n}$ and by (2.6),

(2.8) $\left\lVert
T\right\rVert_{p}^{*}\asymp_{p}N^{r-1/p}\asymp_{p}\left({n}/{s}\right)^{r-1/p}.$

Then it follows from (2.7) that for $|x|\leqslant{\pi}/{(4rN)}\asymp_{p}s/n$
we have

$\displaystyle|T^{(s)}(x)|$ $\displaystyle\geqslant$
$\displaystyle\left|\sum_{m=-rN}^{rN}\widehat{J}_{r,N}(m)(n-rN+m)^{s}\cos
mx\right|$ $\displaystyle\gtrsim$ $\displaystyle\sum_{|m|\leqslant\lambda
N}\widehat{J}_{r,N}(m)(n-rN+m)^{s}$ $\displaystyle\gtrsim_{p}$
$\displaystyle(n-rN)^{s}N^{r}\geqslant(1-1/s)^{s}n^{s}N^{r}\gtrsim_{p}n^{s}\left({n}/{s}\right)^{r}.$

Hence,

(2.9) $\left\lVert
T^{(s)}\right\rVert_{q}^{*}\geqslant\left(\int_{|x|\leqslant\pi/(4rN)}\left|T^{(s)}(x)\right|^{q}dx\right)^{1/q}\gtrsim_{p,q}n^{s}\left({n}/{s}\right)^{r-1/q},$

and combining (2.8) with (2.9), we obtain

(2.10) $\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\ \left\lVert
T_{n}\right\rVert_{p}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}\gtrsim_{p,q}n^{s}\Big{(}{n}/{s}\Big{)}^{{1}/{p}-{1}/{q}}.$

Finally, (2.4) for $n>4rs$ and $s\geqslant 2$ follows from (2.10) and (2.5).
This completes the proof of the lemma. ∎

##  3\. Upper Bounds in Theorem 1.1

In this section, we prove the $\lesssim$ part of Theorem 1.1.

###  3.1. The case $p\in[1,{\infty})$

######  Lemma 3.1.

If $n,s\in\mathbb{N}$ and $r\in(1,\infty],$ then

(3.1) $\left\lVert
D_{n}^{(s)}\right\rVert_{r}^{*}\lesssim_{r}n^{s}\left(1+\left({n}/{s}\right)^{1-{1}/{r}}\right).$

###### Proof.

The inequality is trivial for $r=\infty$. If $r\in(1,{\infty})$, then using
again the Hardy–Littlewood equivalence for monotone Fourier coefficients [20,
Ch. XII, Lemma 6.6], we obtain

(3.2) $\left\lVert
D_{n}^{(s)}\right\rVert_{r}^{*}\lesssim\left\lVert\sum_{k=1}^{n}k^{s}e^{ik\cdot}\right\rVert_{r}^{*}=\left\lVert\sum_{k=1}^{n}(n-k+1)^{s}e^{ik\cdot}\right\rVert_{r}^{*}\lesssim_{r}\left(\sum_{k=1}^{n}k^{r-2}(n-k+1)^{rs}\right)^{{1}/{r}}.$

Next, we have by elementary estimates and by (1.2) and (1.3),

(3.3)
$\displaystyle\sum_{k=3}^{n}\left(\frac{k}{n}\right)^{r-2}\left(1-\frac{k-1}{n}\right)^{rs}\lesssim_{r}n\sum_{k=3}^{n}\int_{(k-2)/n}^{(k-1)/n}t^{r-2}(1-t)^{rs}dt$
$\displaystyle<n\int_{0}^{1}t^{r-2}(1-t)^{rs}dt\lesssim_{r}n{\Gamma(rs+1)}/{\Gamma(rs+r)}\lesssim_{r}ns^{1-r},$

Combining now (3.2) with (3.3), we arrive at

$\displaystyle\left\lVert
D_{n}^{(s)}\right\rVert_{r}^{*}\lesssim_{r}\left(n^{rs}+2^{r-2}(n-1)^{rs}+n^{r-1+rs}s^{1-r}\right)^{1/r}\lesssim_{r}n^{s}+n^{s+1-1/r}s^{1/r-1}.$

Thus (3.1) is established. ∎

As a corollary of Lemma 3.1, we obtain the $\lesssim$ part of Theorem 1.1 for
$p\in[1,{\infty})$.

######  Corollary 3.2.

For $n,s\in\mathbb{N}$ and $1\leqslant p<q\leqslant\infty$,

$\displaystyle\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\ \left\lVert
T_{n}\right\rVert_{p}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}\lesssim_{p,q}n^{s}\left(1+\left({n}/{s}\right)^{{1}/{p}-{1}/{q}}\right).$

###### Proof.

Using Young’s inequality for periodic convolutions and Lemma 3.1, we obtain
for any $T_{n}\in\mathcal{T}_{n}$

$\displaystyle\left\lVert T_{n}^{(s)}\right\rVert_{q}^{*}\lesssim\left\lVert
D_{n}^{(s)}\right\rVert_{r}^{*}\left\lVert
T_{n}\right\rVert_{p}^{*}\lesssim_{p,q}n^{s}\left(1+\left({n}/{s}\right)^{1-{1}/{r}}\right)\left\lVert
T_{n}\right\rVert_{p}^{*},$

where $0\leqslant 1/r:=1/q-1/p+1<1$. ∎

###  3.2. The case $p\in(0,1]$

Here, we prove the $\lesssim$ part of Theorem 1.1 for $p\in(0,1]$. Note that
the case $p=1$ is also discussed in Corollary 3.2. Our main result is the
following:

######  Lemma 3.3.

If $s\in\mathbb{N}$ and $p\in(0,1]$, then for any $f\in E_{p}$,

(3.4) $\left\lVert
f^{(s)}\right\rVert_{\infty}\lesssim\int_{-1}^{1}|\xi|^{s}|\widehat{f}(\xi)|\,d\xi\lesssim_{p}s^{-1/p}\left\lVert
f\right\rVert_{p}.$

######  Remark 3.4.

Note that for $p=1$, the right relation of (3.4) can be easily obtained as
follows:

$\displaystyle\int_{-1}^{1}|\xi|^{s}|\widehat{f}(\xi)|\,d\xi\leqslant\left\lVert
f\right\rVert_{1}\int_{-1}^{1}|\xi|^{s}\,d\xi\lesssim s^{-1}\left\lVert
f\right\rVert_{1}.$

However, the proof for $p\in(0,1)$ is more complicated, and it requires
certain properties of the discrete Hardy spaces.

The counterpart of Lemma 3.3 for trigonometric polynomials is given in the
next result.

######  Lemma 3.5.

If $n,s\in\mathbb{N}$ and $p\in(0,1]$, then for any
$T_{n}(x)=\sum_{k=-n}^{n}c_{k}e^{ikx}\in\mathcal{T}_{n}$,

(3.5) $\left\lVert
T_{n}^{(s)}\right\rVert_{\infty}^{*}\leqslant\sum_{k=-n}^{n}|c_{k}||k|^{s}\lesssim_{p}n^{s}\left(1+\left({n}/{s}\right)^{1/p}\right)\left\lVert
T_{n}\right\rVert_{p}^{*}.$

Combining together Lemmas 3.3 and 3.5 with Bernstein’s inequalities for entire
functions of exponential type $1$ and trigonometric polynomials, we obtain
Theorems 1.2 and 1.1 for $0<p<q\leqslant{\infty}$ and $p\in(0,1]$.

######  Corollary 3.6.

If $s\in\mathbb{N}$ and $0<p<q\leqslant{\infty},\,p\in(0,1]$, then, for any
$f\in E_{p}$,

$\displaystyle\left\lVert
f^{(s)}\right\rVert_{q}\lesssim_{p}s^{{1}/{p}-{1}/{q}}\left\lVert
f\right\rVert_{p}.$

###### Proof.

Using Bernstein’s inequality (see (1.2) for $p=q$) and relation (3.4), we
obtain

$\displaystyle\left\lVert f^{(s)}\right\rVert_{q}\leqslant\left\lVert
f^{(s)}\right\rVert_{\infty}^{1-p/q}\left\lVert
f^{(s)}\right\rVert_{p}^{p/q}\lesssim_{p}s^{{1}/{q}-{1}/{p}}\left\lVert
f\right\rVert_{p}.$

∎

######  Corollary 3.7.

If $n,s\in\mathbb{N}$ and $0<p<q\leqslant\infty,\,p\in(0,1]$, then for any
$T_{n}\in\mathcal{T}_{n}$,

$\displaystyle\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}\lesssim_{p}n^{s}\left(1+\left({n}/{s}\right)^{{1}/{p}-{1}/{q}}\right)\left\lVert
T_{n}\right\rVert_{p}.$

The proof is similar to that of Corollary 3.6.

We first prove Lemma 3.3 for special functions from $E_{p}$.

######  Lemma 3.8.

Let $s\in\mathbb{N}$ and $p\in(0,1]$. In addition, let $a$ be an $H_{p}$-atom
as defined in Definition 7.3, and let $f_{a}$ be defined by (7.1). Then

(3.6)
$\int_{-1}^{1}|\xi|^{s}|\widehat{f_{a}}(\xi)|\,d\xi\lesssim_{p}s^{-1/p}\left\lVert
f_{a}\right\rVert_{p}.$

###### Proof.

Let the interval $I:=[-N+M,N+M],\,N\in\mathbb{N},\,M\in\mathbb{Z}$, contain
the support of $a$. It follows from the properties of the $H_{p}$-atoms (see
Definition 7.3) that $\left\lVert a\right\rVert_{\infty}\lesssim N^{-1/p}$ and
$\sum_{k=-N}^{N}a_{k+M}(k+M)^{j}=0$ for any integer $j$ such that $0\leqslant
j\leqslant p^{-1}-1$. It is easy to show by induction that
$\sum_{k=-N}^{N}a_{k+M}k^{j}=0$ for $0\leqslant j\leqslant p^{-1}-1$.

In addition, $a\in H_{p}(\mathbb{Z})$ by Theorem 7.4. Therefore,
$f_{a}(x)=\sum_{k=-N+M}^{N+M}(-1)^{k}a_{k}\operatorname{sinc}({x-k\pi})$
belongs to $E_{p}$ by Theorem 7.2 and $\left\|f_{a}\right\|_{1}<{\infty}$ by
(1.5). Then we have

$\displaystyle\widehat{f_{a}}(\xi)=(-1)^{M}e^{-iM\pi\xi}\sum_{k=-N}^{N}(-1)^{k}a_{k+M}e^{-ik\pi\xi}\left\\{\begin{array}[]{lll}\pi,&|\xi|<1,\\\
\pi/2,&|\xi|=1,\\\ 0,&|\xi|>1.\end{array}\right.$

Furthermore, setting

$\displaystyle g(\xi):=\pi\sum_{k=-N}^{N}(-1)^{k}a_{k+M}e^{-ik\pi\xi},\qquad
j_{0}:=\lfloor p^{-1}-1\rfloor,$

we observe that the following properties hold:

  1. 1)

$|g(\xi)|=|\widehat{f}_{a}(\xi)|,\,\xi\in(-1,1);$

  2. 2)

$|g(\xi)|\lesssim N^{1-1/p},\,\xi\in\mathbb{R};$

  3. 3)

$g^{(j)}(1)=(-i)^{j}\pi^{j+1}\sum_{k=-N}^{N}a_{k+M}k^{j}=0$ for $0\leqslant
j\leqslant j_{0}$;

  4. 4)

$\left|g^{(j)}(\xi)\right|\lesssim N^{j+1-1/p}$ for $\xi\in\mathbb{R}$ and
$j\geqslant 0$;

  5. 5)

$|g(\xi)|\lesssim(1-\xi)^{j_{0}+1}N^{j_{0}+2-1/p},\,\xi\in[-1,1].$

Note that the last property follows from property 4) and from Taylor’s theorem
for $g$ at $\xi=1$ with the Lagrange form of the remainder since the first
$j_{0}$ derivatives of $g$ vanish at $\xi=1$ by property 3). To conclude the
proof of the lemma, we show that

(3.8) $\int_{0}^{1}\xi^{s}|{g}(\xi)|d\xi\lesssim_{p}s^{-1/p}.$

The estimate

(3.9) $\int_{-1}^{0}|\xi|^{s}|{g}(\xi)|d\xi\lesssim_{p}s^{-1/p}.$

can be established similarly.

If $N>s$, then using property 2), we deduce that

$\displaystyle\int_{0}^{1}\xi^{s}|{g}(\xi)|d\xi\lesssim_{p}N^{1-1/p}s^{-1}\leqslant
s^{-1/p}.$

If $N\leqslant s$, then property 5) and relations (1.2) and (1.3) yield

$\displaystyle\int_{0}^{1}\xi^{s}|{g}(\xi)|d\xi\lesssim
N^{j_{0}+2-1/p}\int_{0}^{1}\xi^{s}(1-\xi)^{j_{0}+1}d\xi$
$\displaystyle\lesssim_{p}$ $\displaystyle
N^{j_{0}+2-1/p}\Gamma(s+1)/\Gamma\left(s+j_{0}+3\right)$
$\displaystyle\lesssim_{p}$ $\displaystyle
N^{j_{0}+2-1/p}s^{-2-j_{0}}\leqslant s^{-1/p}.$

Hence, (3.8) and analogously (3.9) follow. Finally, combining (3.8) and (3.9)
with property 1), we arrive at (3.6). ∎

Having (3.4) for functions $f_{a}$ in hand, we use the atomic structure of the
discrete Hardy spaces given in Theorem 7.4 to extend Lemma 3.8 to any function
$f$.

###### Proof of Lemma 3.3.

By Theorem 7.2, for $f\in E_{p}$ there exists a unique $a\in
H_{p}(\mathbb{Z})$ such that $f=f_{a}$ with $\left\lVert
f\right\rVert_{p}\asymp_{p}\left\lVert a\right\rVert_{H_{p}(\mathbb{Z})}$.
Furthermore, by Theorem 7.4, $a$ has an atomic decomposition
$a=\sum_{d=0}^{\infty}\lambda_{d}a(d)$, where $a(0),\,a(1),\ldots,$ are
$H_{p}$-atoms, with $\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}\asymp_{p}\left\lVert\lambda\right\rVert_{\ell_{p}(\mathbb{Z}_{+})}$.

Then, $f_{a}=\sum_{d=0}^{\infty}\lambda_{d}f_{a(d)}$ and since
$\left\|f_{a}\right\|_{1}<{\infty}$ by (1.5), we also have
$\widehat{f_{a}}=\sum_{d=0}^{\infty}\lambda_{d}\widehat{f}_{a(d)}$. Therefore,
using Lemma 3.8, we conclude that

$\displaystyle\int_{-1}^{1}|\xi|^{s}|\widehat{f}_{a}(\xi)|d\xi$
$\displaystyle\leqslant$
$\displaystyle\int_{-1}^{1}|\xi|^{s}\sum_{d=0}^{\infty}|\lambda_{d}||\widehat{f}_{a(d)}(\xi)|d\xi=\sum_{d=0}^{\infty}|\lambda_{d}|\int_{-1}^{1}|\xi|^{s}|\widehat{f}_{a(d)}(\xi)|d\xi$
$\displaystyle\lesssim_{p}$ $\displaystyle
s^{-1/p}\sum_{d=0}^{\infty}|\lambda_{d}|\leqslant
s^{-1/p}\left\lVert\lambda\right\rVert_{\ell_{p}(\mathbb{Z}_{+})}\lesssim_{p}s^{-1/p}\left\lVert
f\right\rVert_{p}.$

∎

We now use Lemma 3.3 to obtain Lemma 3.5 by a discretization argument.

###### Proof of Lemma 3.5.

Let $T_{n}(x)=\sum_{k=-n}^{n}c_{k}e^{ikx}$ and let $\phi$ be a non-negative
bump function supported in $[-1,1]$ such that $\phi(\tau)=1$ for
$|\tau|\leqslant{1}/{2}$.

If $2n+1\geqslant s$, let us define a function $f$ by the following formula
for its Fourier transform:

(3.10)
$\widehat{f}(\xi):=(2n+1)\sum_{k=-n}^{n}c_{k}\phi\left((2n+1)\xi-2k\right).$

Then $\phi\left((2n+1)\xi-2k\right)$ is supported in
$I_{k}:=\left[\frac{2k-1}{2n+1},\frac{2k+1}{2n+1}\right],\,-n\leqslant
k\leqslant n$, $\widehat{f}$ is supported in $[-1,1]$, and moreover,

(3.11) $\widehat{f}(\xi)=(2n+1)c_{k}\phi\left((2n+1)\xi-2k\right),\qquad\xi\in
I_{k},\quad-n\leqslant k\leqslant n.$

Next, applying the inverse Fourier transform to both sides of (3.10), we
observe that the function

$\displaystyle
f(x)=\check{\phi}\left(\frac{x}{2n+1}\right)\sum_{k=-n}^{n}c_{k}e^{\frac{2ikx}{2n+1}}=\check{\phi}\left(\frac{x}{2n+1}\right)T_{n}\left(\frac{2x}{2n+1}\right)$

belongs to $E_{p}$ and

(3.12) $\left\lVert f\right\rVert_{p}\lesssim_{p}n^{{1}/{p}}\left\lVert
T_{n}\right\rVert_{p}^{*}.$

Indeed,

$\displaystyle\left\lVert
f\right\rVert_{p}=(n+1/2)^{1/p}\left\lVert\Phi_{p}T_{n}\right\rVert_{p}^{*}\leqslant(n+1/2)^{1/p}\left\lVert\Phi_{p}\right\rVert_{\infty}^{*}\left\lVert
T_{n}\right\rVert_{p}^{*},$

where
$\Phi_{p}(y):=\left(\sum_{k\in\mathbb{Z}}\left|\check{\phi}(y/2+k\pi)\right|^{p}\right)^{1/p},\,y\in[-\pi,\pi)$.

Furthermore, using (3.11) and taking into account that $(n+1/2)^{s}\lesssim
n^{s}$ for $2n+1\geqslant s$, we obtain

$\displaystyle\int_{-1}^{1}|\xi|^{s}|\widehat{f}(\xi)|\,d\xi$ $\displaystyle=$
$\displaystyle(2n+1)\int_{-1}^{1}|\xi|^{s}\left|\sum_{k=-n}^{n}c_{k}\phi\left((2n+1)\xi-2k\right)\right|d\xi$
$\displaystyle=$
$\displaystyle(2n+1)\sum_{k=-n}^{n}|c_{k}|\int_{I_{k}}|\xi|^{s}\phi\left((2n+1)\xi-2k\right)d\xi$
$\displaystyle\geqslant$
$\displaystyle(2n+1)\sum_{k=-n}^{n}|c_{k}|\frac{\left|(k+1/4)^{s+1}-(k-1/4)^{s+1}\right|}{(s+1)(n+1/2)^{s+1}}$
$\displaystyle\gtrsim$ $\displaystyle n^{-s}\sum_{k=-n}^{n}|c_{k}||k|^{s}.$

Therefore, using Lemma 3.3 and relations (3.12) and (3.2), we conclude that

(3.14) $\sum_{k=-n}^{n}|c_{k}||k|^{s}\lesssim_{p}n^{s+1/p}s^{-1/p}\left\lVert
T_{n}\right\rVert_{p}^{*},\qquad 2n+1\geqslant s.$

If $2n+1<s$ and $s\geqslant 2$, we proceed by using a modification of the
previous argument. Let

$\displaystyle\widehat{f}(\xi):=s\sum_{k=-n}^{n}c_{k}\phi\left(s\xi-(s-1){k}/{n}\right).$

Then $\phi\left(s\xi-(s-1){k}/{n}\right)$ is supported in
$I_{k}:=\left[-\frac{1}{s}+\frac{(s-1)k}{ns},\frac{1}{s}+\frac{(s-1)k}{ns}\right]$,
$-n\leqslant k\leqslant n$, $\widehat{f}$ is supported in $[-1,1]$, and
moreover,

(3.15) $\widehat{f}(\xi)=sc_{k}\phi\left(s\xi-(s-1){k}/{n}\right),\qquad\xi\in
I_{k},\quad-n\leqslant k\leqslant n,$

since the family $\left\\{I_{k}\right\\}_{k=-n}^{n}$ is pairwise disjoint by
the assumption $2n+1<s$. Next, the function

$\displaystyle
f(x)=\check{\phi}(x/s)\sum_{k=-n}^{n}c_{k}e^{(1-s^{-1})\frac{ikx}{n}}=\check{\phi}\left(\frac{x}{s}\right)T_{n}\left(\frac{\left(1-s^{-1}\right)x}{n}\right)$

belongs to $E_{p}$ and

(3.16) $\left\lVert f\right\rVert_{p}\lesssim_{p}s^{{1}/{p}}\left\lVert
T_{n}\right\rVert_{p}^{*}.$

Indeed, note that the polynomial $T_{n}\left((2\pi/\alpha)x\right)$ is
$\alpha$-periodic with $\alpha:={2\pi n}/{(1-s^{-1})}$, and since
$\check{\phi}$ is rapidly decaying, it satisfies the estimate
$\left|\check{\phi}(x/s)\right|\lesssim_{p}{(1+|x/s|)^{-2/p}}$. Then we have

$\displaystyle\left\lVert f\right\rVert_{p}$ $\displaystyle\lesssim_{p}$
$\displaystyle\left(\int_{\mathbb{R}}(1+|x/s|)^{-2}\left|T_{n}((2\pi/\alpha)x)\right|^{p}\right)^{1/p}$
$\displaystyle\leqslant$
$\displaystyle\left(\sum_{j=0}^{\infty}\frac{1}{(1+j)^{2}}\int_{js}^{(j+1)s}\left(\left|T_{n}((2\pi/\alpha)x)\right|^{p}+\left|T_{n}(-(2\pi/\alpha)x)\right|^{p}\right)dx\right)^{1/p}$
$\displaystyle\leqslant$
$\displaystyle\left(\sum_{j=0}^{\infty}\frac{1}{(1+j)^{2}}\int_{\alpha\lfloor
js/\alpha\rfloor}^{\alpha\lceil(j+1)s/\alpha\rceil}\left(\left|T_{n}((2\pi/\alpha)x)\right|^{p}+\left|T_{n}(-(2\pi/\alpha)x)\right|^{p}\right)dx\right)^{1/p}$
$\displaystyle=$
$\displaystyle\left(\sum_{j=0}^{\infty}\frac{\alpha}{(1+j)^{2}}\int_{\lfloor
js/\alpha\rfloor}^{\lceil(j+1)s/\alpha\rceil}\left(\left|T_{n}(2\pi
x)\right|^{p}+\left|T_{n}(-2\pi x)\right|^{p}\right)dx\right)^{1/p}$
$\displaystyle\lesssim_{p}$
$\displaystyle\left(\sum_{j=0}^{\infty}\frac{1}{(1+j)^{2}}\right)^{1/p}\left((2\alpha+s)\int_{0}^{1}\left|T_{n}(2\pi
x)\right|^{p}dx\right)^{1/p}.$

Since $\alpha<\pi s$, we see that (3.2) implies (3.16). Finally, using (3.15),
we obtain

$\displaystyle\int_{-1}^{1}|\xi|^{s}|\widehat{f}(\xi)|\,d\xi$ $\displaystyle=$
$\displaystyle
s\int_{-1}^{1}|\xi|^{s}\left|\sum_{k=-n}^{n}c_{k}\phi\left(s(\xi-(1-s^{-1}){k}/{n})\right)\right|d\xi$
$\displaystyle=$ $\displaystyle
s\sum_{k=-n}^{n}|c_{k}|\int_{I_{k}}|\xi|^{s}\phi\left(s(\xi-(1-s^{-1}){k}/{n})\right)d\xi$
$\displaystyle\geqslant$
$\displaystyle\frac{s}{s+1}\sum_{k=-n}^{n}|c_{k}|\left|\left(\frac{(s-1)k}{sn}+\frac{1}{2s}\right)^{s+1}-\left(\frac{(s-1)k}{sn}-\frac{1}{2s}\right)^{s+1}\right|$
$\displaystyle\geqslant$
$\displaystyle\sum_{k=-n}^{n}|c_{k}|\left((1-s^{-1}){|k|}/{n}\right)^{s}\gtrsim
n^{-s}\sum_{k=-n}^{n}|c_{k}||k|^{s}.$

Therefore, using Lemma 3.3 and relations (3.16) and (3.2), we conclude that

(3.19) $\sum_{k=-n}^{n}|c_{k}||k|^{s}\lesssim_{p}n^{s}\left\lVert
T_{n}\right\rVert_{p}^{*},\qquad 2n+1<s.$

Thus, the right relation of (3.5) follows from (3.14) and (3.19). Lemma 3.5
and Corollary 3.7 are established. ∎

##  4\. Properties of Concave Sequences and Polynomials from
$\mathcal{T}_{n,\mbox{c}}$

Throughout this section, $c=(c_{0},\ldots,c_{n})$ is a concave sequence (see
Definition 1.3). The proof of Theorem 1.4 is based on several technical lemmas
and on Proposition 4.5, all of which are of independent interest.

Let us define the following sequences:

(4.1) $\displaystyle
v_{l}(k):=\min\left\\{1-\frac{k-1}{n},1-\frac{l-1}{n}\right\\},\qquad
l=0,\ldots,n,\quad k=0,\ldots,n;$ $\displaystyle
V_{l}:=(v_{l}(0),\ldots,v_{l}(n)),\qquad l=0,\ldots,n.$

The sequences $V_{l},\,l=0,\ldots,n$, are concave. Moreover, it turns out that
every concave sequence can be represented as a linear combination of
$V_{l},\,l=0,\ldots,n$, with nonnegative coefficients.

######  Lemma 4.1.

If $c$ is concave, then there exist numbers $h_{l}\geqslant 0,\,l=0,\ldots,n$,
such that $c=\sum_{l=0}^{n}h_{l}V_{l}$.

###### Proof.

Since $c_{k}=\sum_{j=k}^{n}\Delta c_{j}$ with $c_{n+1}:=0$ and $\Delta
c_{j}=\Delta c_{0}+\sum_{l=1}^{j}\left(-\Delta^{2}c_{l-1}\right)$, we have
$c_{k}=\sum_{j=k}^{n}\left(\Delta
c_{0}+\sum_{l=1}^{j}\left(-\Delta^{2}c_{l-1}\right)\right),\,0\leqslant
k,j\leqslant n$. Hence for $0\leqslant k\leqslant n$,

$\displaystyle c_{k}=\Delta
c_{0}\,(n-k+1)+\sum_{l=1}^{n}\left(-\Delta^{2}c_{l-1}\right)\min\\{n-k+1,n-l+1\\}:=n\sum_{l=0}^{n}h_{l}v_{l}(k),$

since $n-k+1=\min\\{n-k+1,n+1\\}=nv_{0}(k)$. ∎

The next three lemmas contain technical estimates for concave sequences.

######  Lemma 4.2.

The following relations are valid:

(4.2)
$\sum_{k=0}^{n}\frac{v_{l}(k)}{n-k+1}\asymp\left(1-\frac{l-1}{n}\right)\left(\left|\log\left(1-\frac{l-1}{n}\right)\right|+1\right),\qquad
0\leqslant l\leqslant n;$ (4.3) $\min_{0\leqslant l\leqslant
n-1}\left(1-\frac{l}{n}\right)\left(\left|\log\left(1-\frac{l}{n}\right)\right|+1\right)=\frac{\log
n+1}{n}.$

###### Proof.

First, we note that the function $y(\log(1/y)+1)$ is increasing on $(0,1)$,
which yields (4.3). Next, by the definition of $v_{l}(k)$,

(4.4)
$\sum_{k=0}^{n}\frac{v_{l}(k)}{n-k+1}=\left(1-\frac{l-1}{n}\right)\left(\sum_{m=n-l+1}^{n+1}\frac{1}{m}+1\right)-\frac{1}{n},\qquad
0\leqslant l\leqslant n.$

Hence, (4.2) is valid for $l=0$. In addition,

(4.5)
$\sum_{m=n-l+1}^{n+1}\frac{1}{m}+1\asymp\left|\log\left(1-\frac{l-1}{n}\right)\right|+1,\qquad
1\leqslant l\leqslant n.$

Thus, (4.2) follows from relations (4.3)–(4.5). ∎

######  Lemma 4.3.

For $n,s\in\mathbb{N}$ we have

(4.6)
$\sup_{\tau\in[0,1]}H_{n,s}(\tau):=\sup_{\tau\in[0,1]}\frac{\tau^{s+1}}{1-\log\left(1+n^{-1}-\tau\right)}\lesssim\frac{1}{\log(s+2)}+\frac{1}{\log(n+2)}.$

###### Proof.

Let us set ${\gamma}(s):=(s+2)^{-1}\log\log(s+2)$. First, we have

(4.7)
$H_{n,s}(\tau)\lesssim\tau^{s+1}\leqslant\frac{1}{\log(s+2)},\qquad\tau\in\left[0,1-{\gamma}(s)\right.\left.\right).$

Next, for $\tau\in\left[1-{\gamma}(s),1\right]$ we obtain

$\displaystyle 1-\log(1+n^{-1}-\tau)\geqslant
1-\log\left(s^{-1}\log\log(s+2)+n^{-1}\right)$ $\displaystyle\geqslant$
$\displaystyle 1-\log\left((s+2)^{-1/2}+n^{-1/2}\right)$
$\displaystyle\gtrsim$ $\displaystyle\log(\min\\{n+2,s+2\\}).$

Hence

(4.8)
$H_{n,s}(\tau)\lesssim\frac{1}{1-\log(1+n^{-1}-\tau)}\lesssim\frac{1}{\log(s+2)}+\frac{1}{\log(n+2)},\qquad\tau\in\left[1-{\gamma}(s),1\right].$

Thus, the upper estimate in (4.6) follows from (4.7) and (4.8). ∎

The next lemma discusses a relation between concave sequences.

######  Lemma 4.4.

If $c$ is concave, then

(4.9) $\frac{1}{n}\sum_{k=0}^{n}c_{k}+\int_{n^{-1}}^{\pi}x^{-1}c_{n-{\lfloor
x^{-1}+1\rfloor}}dx\lesssim\sum_{k=0}^{n}\frac{c_{k}}{n-k+1}.$

###### Proof.

By Lemma 4.1, it suffices to establish (4.9) for all
$c_{k}=v_{l}(k),\,0\leqslant k\leqslant n$, defined by (4.1), with a fixed
$l,\,0\leqslant l\leqslant n$. In addition, taking into account (4.2), we see
that inequality (4.9) is a consequence of the following inequality:

(4.10)
$\frac{1}{n}{\sum_{k=0}^{n}v_{l}(k)}+\int_{n^{-1}}^{\pi}x^{-1}v_{l}({n-{\lfloor
x^{-1}+1\rfloor})}dx\lesssim\left(1-\frac{l-1}{n}\right)\left(\left|\log\left(1-\frac{l-1}{n}\right)\right|+1\right).$

To prove (4.10), we first note that the estimate

(4.11) $\frac{1}{n}{\sum_{k=0}^{n}v_{l}(k)}\leqslant
2\left(1-\frac{l-1}{n}\right),\qquad 0\leqslant l\leqslant n,$

immediately follows from (4.1). Next, we show that for $0\leqslant l\leqslant
n$,

(4.12) $I_{n}(l):=\int_{n^{-1}}^{\pi}x^{-1}v_{l}({n-{\lfloor
x^{-1}+1\rfloor})}dx\lesssim\left(1-\frac{l-1}{n}\right)\left(\left|\log\left(1-\frac{l-1}{n}\right)\right|+1\right).$

To prove it, we need the following simple estimates for $x\in(0,\pi]$:

(4.13) $v_{l}({n-{\lfloor
x^{-1}+1\rfloor}})\leqslant\min\left\\{\frac{2}{n}+\frac{1}{nx},1-\frac{l-1}{n}\right\\}\leqslant\min\left\\{\frac{2\pi+1}{nx},1-\frac{l-1}{n}\right\\}.$

If $l=0$, then using (4.13), we see that $I_{n}(0)\lesssim 1$, that is, (4.12)
holds. Furthermore, using again (4.13) for $1\leqslant l\leqslant n$, we have

$\displaystyle I_{n}(l)$ $\displaystyle=$
$\displaystyle\int_{n^{-1}}^{n^{-1}\left(1-\frac{l-1}{n}\right)^{-1}}x^{-1}v_{l}({n-{\lfloor
x^{-1}+1\rfloor}})dx$ $\displaystyle+$
$\displaystyle\int_{{n^{-1}\left(1-\frac{l-1}{n}\right)^{-1}}}^{\pi}x^{-1}v_{l}({n-{\lfloor
x^{-1}+1\rfloor}})dx$ $\displaystyle\lesssim$
$\displaystyle\left(1-\frac{l-1}{n}\right)\int_{n^{-1}}^{n^{-1}\left(1-\frac{l-1}{n}\right)^{-1}}x^{-1}dx+\frac{1}{n}\int_{{n^{-1}\left(1-\frac{l-1}{n}\right)^{-1}}}^{\pi}x^{-2}dx$
$\displaystyle\lesssim$
$\displaystyle\left(1-\frac{l-1}{n}\right)\left(\left|\log(1-\frac{l-1}{n})\right|+1\right).$

Therefore, (4.12) holds for $0\leqslant l\leqslant n$, and (4.10) follows from
(4.11) and (4.12), completing the proof. ∎

Our next result describes the integral norm of a trigonometric polynomial with
concave coefficients.

######  Proposition 4.5.

Let $c$ be concave and $T_{n}(x)=c_{0}+2\sum_{k=1}^{n}c_{k}\cos{kx}$. Then we
have

(4.14) $\left\lVert
T_{n}\right\rVert_{1}^{*}\asymp\sum_{k=0}^{n}\frac{c_{k}}{n-k+1}.$

The proof is based on two lemmas. The first one is a Hardy-type estimate (cf.,
e.g, [20, Theorem 8.7]).

######  Lemma 4.6.

For any $T_{n}(x)=c_{0}+2\sum_{k=1}^{n}c_{k}\cos{kx}$, we have

(4.15) $\left\lVert
T_{n}\right\rVert_{1}^{*}\gtrsim\left|\sum_{k=0}^{n}\frac{c_{k}}{n-k+1}\right|.$

###### Proof.

For $c_{0}=1$ and positive coefficients a weaker version of inequality (4.15)
was proved by Nikolskii [15, Theorem 1]. The proof of (4.15) is similar to
that one. Since the polynomial

$\displaystyle Q_{2n+1}(x)$ $\displaystyle:=$
$\displaystyle\frac{1}{2(n+1)}+2\sin\,(n+1)x\sum_{k=1}^{n}\frac{\sin kx}{k}$
$\displaystyle=$ $\displaystyle\frac{1}{2(n+1)}+\sum_{k=1}^{n}\frac{\cos
kx}{n-k+1}-\sum_{k=1}^{n}\frac{\cos\,(k+n+1)x}{k}$

is uniformly bounded on $[-\pi,\pi)$ (cf. [20, Ch. VIII, Eqns. (1.4) and
(1.6)]), we obtain

$\displaystyle\left\lVert
T_{n}\right\rVert_{1}^{*}\gtrsim\left|\int_{-\pi}^{\pi}T_{n}(x)Q_{2n+1}(x)dx\right|\gtrsim\left|\sum_{k=0}^{n}\frac{c_{k}}{n-k+1}\right|.$

∎

######  Lemma 4.7.

If $c$ is concave and $T_{n}(x)=c_{0}+2\sum_{k=1}^{n}c_{k}\cos{kx}$, then

(4.16) $\displaystyle|T_{n}(x)|\lesssim x^{-1}c_{n-{\lfloor
x^{-1}+1\rfloor}}\qquad x\in(1/n,\pi),$ (4.17) $\displaystyle\left\lVert
T_{n}\right\rVert_{1}^{*}\lesssim\frac{1}{n}\sum_{k=0}^{n}c_{k}+\int_{n^{-1}}^{\pi}x^{-1}c_{n-{\lfloor
x^{-1}+1\rfloor}}dx.$

###### Proof.

We prove the lemma by the standard argument (see, e.g., [20, Ch. V, Sect. 1]).
Since $T_{n}$ is a polynomial with non-increasing coefficients, the Abel
transformation shows that for $x\in(1/n,\pi)$,

$\displaystyle|T_{n}(x)|$ $\displaystyle\lesssim$ $\displaystyle
x^{-1}\left|\sum_{k=0}^{n}\Delta c_{k}\sin{(k+1/2)x}\right|$
$\displaystyle\leqslant$ $\displaystyle x^{-1}\left|\sum_{k=0}^{n}\Delta
c_{k}e^{ikx}\right|+x^{-1}\left|\sum_{k=0}^{n}\Delta
c_{k}e^{-ikx}\right|=:I_{1}(x)+I_{2}(x),$

where $c_{n+1}:=0$. Hence, setting $b_{k}:=\Delta c_{n-k},\,1\leqslant
k\leqslant n,$ we obtain

$\displaystyle I_{1}(x)=x^{-1}\left|\sum_{k=0}^{n}b_{k}e^{-ikx}\right|\lesssim
x^{-1}\sum_{k=0}^{\lfloor x^{-1}+1\rfloor}b_{k}+x^{-1}\left|\sum_{k=\lfloor
x^{-1}+1\rfloor}^{n}b_{k}e^{-ikx}\right|.$

Now, applying again the Abel transformation to the sum $\sum_{k=\lfloor
x^{-1}+1\rfloor}^{n}$ above and using the fact that $b_{k}$ is non-increasing
with $k$, we arrive at

(4.19) $I_{1}(x)\lesssim x^{-1}\sum_{k=0}^{\lfloor
x^{-1}+1\rfloor}b_{k}+x^{-2}b_{\lfloor x^{-1}+1\rfloor}\lesssim
x^{-1}\sum_{k=0}^{\lfloor x^{-1}+1\rfloor}b_{k}=x^{-1}c_{n-{\lfloor
x^{-1}+1\rfloor}}.$

The similar estimate holds for $I_{2}(x)$. Thus, (4.16) follows from (4),
(4.19), and the corresponding estimate for $I_{2}(x)$. To prove (4.17), we
split the integral
$\int_{0}^{\pi}|T_{n}(x)|dx\leqslant(1/n)\|T_{n}\|_{\infty}^{*}+\int_{1/n}^{\pi}|T_{n}(x)|dx$
and use (4.16). ∎

###### Proof of Proposition 4.5..

The $\lesssim$ part of relation (4.14) is a consequence of (4.9) and (4.17),
while the $\gtrsim$ part immediately follows from (4.15). ∎

######  Remark 4.8.

A weaker version of the $\lesssim$ part of Proposition 4.5 is a special case
of the following Efimov result [8, Theorem 1] with a lengthy proof: for any
$T_{n}(x)=\sum_{k=0}^{n}c_{k}\cos{kx}$,

$\displaystyle\left\lVert
T_{n}\right\rVert_{1}^{*}\lesssim\left|c_{0}\right|+\frac{1}{n+1}\sum_{k=0}^{n-1}{(k+1)(n-k)}\left|\Delta^{2}c_{k}\right|+\sum_{k=0}^{n}\frac{|c_{k}|}{n-k+1}.$

Note that for concave coefficients,
$\left|\Delta^{2}c_{k}\right|=-\Delta^{2}c_{k}$ and

$\displaystyle\frac{1}{n+1}\sum_{k=0}^{n-1}{(k+1)(n-k)}\left|\Delta^{2}c_{k}\right|=\frac{2}{n+1}\sum_{k=0}^{n}c_{k}\leqslant\sum_{k=0}^{n}\frac{c_{k}}{n-k+1}.$

##  5\. Proofs of Theorems

Theorem 1.1 immediately follows from Lemma 2.2 and Corollaries 3.2 and 3.7. We
now prove Theorem 1.2.

###### Proof of Theorem 1.2.

The $\gtrsim$ part follows from Lemma 2.1. To prove the $\lesssim$ part of the
theorem, we need the following result:

######  Lemma 5.1.

(Ganzburg and Tikhonov [11, Theorem 1.4]) For $0<p<q\leqslant\infty$ and
$s=0,\,1,\ldots$, the following relation holds:

$\displaystyle\sup_{\begin{subarray}{c}f\in E_{p},\\\ \left\lVert
f\right\rVert_{p}=1\end{subarray}}\left\lVert
f^{(s)}\right\rVert_{q}\leqslant\liminf_{n\to{\infty}}n^{-s-1/p+1/q}\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\
\left\lVert T_{n}\right\rVert_{p}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{q}^{*}.$

This lemma allows us to derive the $\lesssim$ part of Theorem 1.2 from the
$\lesssim$ part of Theorem 1.1. The proof of the theorem is complete. ∎

Note that the $\lesssim$ part of the theorem for $p\in(0,1]$ is proved in
Corollary 3.6 by a different method.

###### Proof of Theorem 1.4.

For a polynomial
$T_{n}(\cdot)=c_{0}+2\sum_{k=1}^{n}c_{k}\cos{k\cdot}\in\mathcal{T}_{n,\mbox{c}}$,
we show that

(5.1)
$\sum_{k=0}^{n}k^{s}c_{k}\lesssim\left(n^{s}+\frac{n^{s+1}}{s+1}\right)\left(\frac{1}{\log{(s+2)}}+\frac{1}{\log(n+2)}\right)\sum_{k=0}^{n}\frac{c_{k}}{n-k+1}.$

Then the $\lesssim$ part of (1.8) follows from relation (5.1) and Proposition
4.5. It remains to verify (5.1).

By Lemma 4.1, it suffices to establish (5.1) for all
$c_{k}=v_{l}(k),\,0\leqslant k\leqslant n$, with a fixed $l,\,0\leqslant
l\leqslant n$. Recall that by relation (4.2) of Lemma 4.2,

(5.2)
$\left|V_{l}\right|_{1}:=\sum_{k=0}^{n}\frac{v_{l}(k)}{n-k+1}\asymp\left(1-\frac{l-1}{n}\right)\left(\left|\log\left(1-\frac{l-1}{n}\right)\right|+1\right).$

Then it follows from (5.1) and (5.2) that it suffices to prove the following
relation with a fixed $l,\,0\leqslant l\leqslant n$:

(5.3)
$\sum_{k=0}^{n}k^{s}v_{l}(k)\lesssim\left(n^{s}+\frac{n^{s+1}}{s+1}\right)\left(\frac{1}{\log{(s+2)}}+\frac{1}{\log(n+2)}\right)\left|V_{l}\right|_{1}.$

To prove (5.3), we use the definition of $v_{l}(k)$ (see (4.1)) and split the
left-hand side of (5.3) into two sums as follows:

(5.4)
$\sum_{k=0}^{n}k^{s}v_{l}(k)=\left(1-\frac{l-1}{n}\right)\sum_{k=0}^{l-1}k^{s}+\sum_{k=l}^{n}k^{s}\left(1-\frac{k-1}{n}\right)=:I_{1}(l)+I_{2}(l).$

Next, $I_{1}(0)=0$ and using (5.2) and Lemma 4.3 for $n,s\in\mathbb{N}$ and
$0\leqslant l\leqslant n$, we obtain

$\displaystyle I_{1}(l)$ $\displaystyle\leqslant$
$\displaystyle\left(1-\frac{l-1}{n}\right)\frac{l^{s+1}}{s+1}\lesssim\frac{\left|V_{l}\right|_{1}}{s+1}\left(\frac{l^{s+1}}{1-\log\left(1-\frac{l-1}{n}\right)}\right)$
$\displaystyle=$
$\displaystyle\frac{n^{s+1}\left|V_{l}\right|_{1}}{s+1}H_{n,s}(l/n)\lesssim\frac{n^{s+1}}{s+1}\left(\frac{1}{\log(s+2)}+\frac{1}{\log(n+2)}\right)\left|V_{l}\right|_{1},$

where $H_{n,s}$ is defined by (4.6). To estimate $I_{2}(l)$, note that

(5.6)
$I_{2}(l)=n^{s-1}+\frac{2}{n}\sum_{k=l}^{n-1}k^{s}+\sum_{k=l}^{n-1}k^{s}\left(1-\frac{k+1}{n}\right)\leqslant
n^{s-1}+\frac{2n^{s}}{s+1}+\sum_{k=l}^{n-1}k^{s}\left(1-\frac{k+1}{n}\right).$

Furthermore, for $0\leqslant l\leqslant n-1$,

(5.7)
$\displaystyle\sum_{k=l}^{n-1}k^{s}\left(1-\frac{k+1}{n}\right)<n^{s+1}\int\limits_{l/n}^{1}t^{s}(1-t)dt$
$\displaystyle=n^{s+1}\left(\frac{1}{s+1}-\frac{\left({l}/{n}\right)^{s+1}}{s+1}-\frac{1}{s+2}+\frac{\left({l}/{n}\right)^{s+2}}{s+2}\right)<n^{s+1}\frac{1-\left(({l-1})/{n}\right)^{s+1}}{s(s+1)}.$

In addition, we need the following elementary inequality:

(5.8)
$\frac{1-x^{s+1}}{s+1}\leqslant\frac{(1-x)\left(\log\frac{1}{1-x}+1\right)}{\log
s},\qquad x\in[0,1).$

To prove it, we first note that $(1-x^{s+1})/(s+1)\leqslant 1-x$ and, in
addition, inequalities $x>1-1/s$ and $\log(1/(1-x))>\log s$ are equivalent.
Hence (5.8) is valid for $x\in(1-1/s,1)$. Next, since the function
$y(\log(1/y)+1)$ is increasing on $[0,1)$, we have

$\displaystyle\frac{(1-x)\left(\log\frac{1}{1-x}+1\right)}{\log
s}>\frac{1}{s}>\frac{1-x^{s+1}}{s+1},\qquad x\in[0,1-1/s].$

Hence, inequality (5.8) is established. Then combining (5.7) and (5.2) with
(5.8) for $x=(l-1)/n$, we obtain

(5.9)
$\sum_{k=l}^{n-1}k^{s}\left(1-\frac{k+1}{n}\right)\leqslant\frac{n^{s+1}}{s\log
s}\left(1-\frac{l-1}{n}\right)\left(\left|\log\left(1-\frac{l-1}{n}\right)+1\right|\right)\lesssim\frac{n^{s+1}}{s\log
s}\left|V_{l}\right|_{1}.$

The following inequalities are valid as well by relations (5.2) and (4.3):

(5.10)
${n^{s-1}}\lesssim\frac{n^{s}}{\log(n+2)}\left|V_{l}\right|_{1},\qquad\frac{n^{s}}{s}\lesssim\frac{n^{s+1}}{s\log(n+2)}\left|V_{l}\right|_{1}.$

Therefore, it follows from (5.6), (5.9), and (5.10) that

(5.11)
$I_{2}(l)\lesssim\left(n^{s}+\frac{n^{s+1}}{s+1}\right)\left(\frac{1}{\log{(s+2)}}+\frac{1}{\log(n+2)}\right)\left|V_{l}\right|_{1}.$

Thus, collecting together (5.4), (5), and (5.11), we arrive at (5.3),
eventually proving (5.1) and, therefore, the $\lesssim$ part of Theorem 1.4.

To establish the $\gtrsim$ part of the theorem, we choose
$T_{n,l}(\cdot):=v_{l}(0)+2\sum_{k=1}^{n}v_{l}(k)\cos
k\cdot\in\mathcal{T}_{n,\mbox{c}}$ with a certain $l,\,0\leqslant l\leqslant
n$. Recall that the concave sequence $V_{l}=(v_{l}(0),\ldots,v_{l}(n))$ is
defined by (4.1).

We first assume that $s\in\mathbb{N}$ is an even number. In the case of
$1\leqslant n\leqslant s$, let us set $l=n$. Then by relations (4.1) and (5.2)
and, in addition, by Proposition 4.5, we obtain

$\displaystyle\left\lVert T_{n,n}^{(s)}\right\rVert_{{\infty}}^{*}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}k^{s}v_{n}(k)>n^{s}v_{n}(n)=n^{s-1}\gtrsim\frac{n^{s}}{\log(n+2)}\left|V_{n}\right|_{1}$
$\displaystyle\gtrsim$
$\displaystyle\left(n^{s}+n^{s+1}s^{-1}\right)\left(\frac{1}{\log(s+2)}+\frac{1}{\log(n+2)}\right)\left\lVert
T_{n,n}\right\rVert_{1}^{*}.$

In the case of $n>s\geqslant 1$, let us set $l=1+\lfloor n(1-1/s)\rfloor$.
Then

(5.13) $1\leqslant l<n,\qquad l^{s+1}\gtrsim n^{s+1},\qquad 1-(l-1)/n\geqslant
1/s.$

Therefore, using relations (4.1), (5.2), and (5.13) and, in addition, using
Proposition 4.5, we obtain

$\displaystyle\left\lVert T_{n,l}^{(s)}\right\rVert_{{\infty}}^{*}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}k^{s}v_{l}(k)\geqslant\sum_{k=1}^{l}k^{s}v_{l}(k)$
$\displaystyle\gtrsim$
$\displaystyle\frac{\left(1-\frac{l-1}{n}\right)\frac{{l}^{s+1}}{s}}{\left(1-\frac{{l}-1}{n}\right)\left(\left(\left|\log(1-\frac{{l}-1}{n}\right)\right|+1\right)}\left|V_{l}\right|_{1}\gtrsim\frac{n^{s+1}}{s\log(s+1)}\left|V_{l}\right|_{1}$
$\displaystyle\gtrsim$
$\displaystyle\left(n^{s+1}s^{-1}+n^{s}\right)\left(\frac{1}{\log(s+2)}+\frac{1}{\log(n+2)}\right)\left\lVert
T_{n,l}\right\rVert_{1}^{*}.$

Thus, for an even $s$, the $\gtrsim$ part of Theorem 1.4 follows from (5) and
(5).

Finally, for an odd $s$, using Bernstein’s inequality $\left\lVert
T_{n}^{(s+1)}\right\rVert_{\infty}^{*}\leqslant n\left\lVert
T_{n}^{(s)}\right\rVert_{\infty}^{*}$, we conclude that the following
relations hold:

$\displaystyle\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n,\mbox{c}},\\\
\left\lVert T_{n}\right\rVert_{1}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s)}\right\rVert_{\infty}^{*}$ $\displaystyle\geqslant$ $\displaystyle
n^{-1}\sup_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n,\mbox{c}},\\\
\left\lVert T_{n}\right\rVert_{1}^{*}=1\end{subarray}}\left\lVert
T_{n}^{(s+1)}\right\rVert_{\infty}^{*}$ $\displaystyle\gtrsim$
$\displaystyle\left(n^{s}+\frac{n^{s+1}}{s+1}\right)\left(\frac{1}{\log{(s+2)}}+\frac{1}{\log(n+2)}\right).$

The proof is now complete. ∎

##  6\. Remarks on Extremal Polynomials

Below we discuss several facts about the extremal trigonometric polynomials
$P_{n}\in\mathcal{T}_{n}$ for Bernstein–Nikolskii inequality (1.4), with
$p=1,\,q=\infty$, and $s=0,\,1,\ldots$. In the special case of $s=0$, the
similar results are known either for trigonometric polynomials or entire
functions of exponential type (cf. [13, 19, 2]).

The polynomials $P_{n}$ are defined by the relations

(6.1)
$P_{n}^{(s)}(0)=1,\qquad\inf_{\begin{subarray}{c}T_{n}\in\mathcal{T}_{n},\\\
T_{n}^{(s)}(0)=1\end{subarray}}\left\lVert
T_{n}\right\rVert_{1}^{*}=\left\|P_{n}\right\|_{1}^{*},$

and their existence is well known. In addition, it is clear that by
considering $\left(\overline{P_{n}(x)}+P_{n}(x)\right)/2$ and
$\left(P_{n}(x)+(-1)^{s}P_{n}(-x)\right)/2$, we may assume that first, $P_{n}$
is real-valued, and second, it is even for even $s$ and odd for odd $s$.

######  Lemma 6.1.

Let $P_{n}$ satisfy (6.1). Then for any $Q_{n}\in\mathcal{T}_{n}$, we have

(6.2)
$\int_{-\pi}^{\pi}\operatorname{sign}\left(P_{n}(x)\right)\,Q_{n}(x)dx=\left\lVert
P_{n}\right\rVert_{1}^{*}\,Q_{n}^{(s)}(0).$

###### Proof.

For any $Q_{n}\in\mathcal{T}_{n}$, the function

$\displaystyle{\varphi}({\lambda}):=\left\lVert
R_{n,{\lambda}}\right\rVert_{1}^{*}:=\left\lVert
P_{n}+\lambda\left(Q_{n}-Q_{n}^{(s)}(0)P_{n}\right)\right\rVert_{1}^{*},\qquad{\lambda}\in\mathbb{R},$

has a minimum at $\lambda=0$ since $R_{n,{\lambda}}\in\mathcal{T}_{n}$ and
$R_{n,{\lambda}}^{(s)}(0)=1$. Hence, calculating
${\varphi}^{\prime}({\lambda})$, we arrive at (6.2). ∎

For $s=0$ and entire functions of exponential type, the similar fact was
proved in [13, Eqn. (2.10)]. As a simple corollary of Lemma 6.1, we obtain the
following result (cf. [19, p. 211] for $s=0$):

######  Corollary 6.2.

Let $P_{n}$ satisfy (6.1). Then the best constant $\left(\left\lVert
P_{n}\right\rVert^{*}_{1}\right)^{-1}$ in Bernstein–Nikolskii inequality
(1.4), with $p=1,\,q=\infty$, and $s=0,\,1,\ldots$, is the distance in the
uniform norm from the function $h_{n,s}:=(1/(2\pi))D_{n}^{(s)}$ to the space
$F_{n}$ of all functions with Fourier coefficients $c_{k}=0$ for
$\left|k\right|\leqslant n$.

###### Proof.

Let us set $\psi_{n}:=\operatorname{sign}(P_{n})\left(\left\lVert
P_{n}\right\rVert^{*}_{1}\right)^{-1}$. It follows from Lemma 6.1 that the
function $h_{n,s}(x)-\psi_{n}(-x)$ belongs to $F_{n}$. Hence,

$\displaystyle\operatorname{dist}_{\infty}(h_{n,s},F_{n})\leqslant\left\lVert\psi_{n}\right\rVert_{\infty}^{*}=\left(\left\lVert
P_{n}\right\rVert^{*}_{1}\right)^{-1}.$

Conversely, for any $f\in F_{n}$,

$\displaystyle\left\lVert
h_{n,s}-f\right\rVert_{\infty}^{*}\geqslant\left(\left\lVert
P_{n}\right\rVert_{1}^{*}\right)^{-1}\int_{-\pi}^{\pi}\left(h_{n,s}(-x)-f(-x)\right)\,P_{n}(x)dx=P_{n}^{(s)}(0)\left(\left\lVert
P_{n}\right\rVert_{1}^{*}\right)^{-1},$

and the result follows. ∎

Finally, we discuss certain properties of zeros of extremal polynomials.

######  Corollary 6.3.

Let $P_{n}$ be a polynomial, satisfying (6.1) for $s=0$. Then the following
properties are valid:

  * (a)

$P_{n}$ has $2n$ simple zeros
$-\pi<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{2n}<\pi$;

  * (b)

for any polynomial $Q_{n}\in\mathcal{T}_{n}$, the following equality holds:

(6.3)
$2\operatorname{sign}(P_{n}(\pi))\sum_{k=1}^{2n}(-1)^{k+1}Q_{n}(\alpha_{k})=\left\lVert
P_{n}\right\rVert_{1}^{*}Q_{n}^{{}^{\prime}}(0).$

###### Proof.

First we recall that $P_{n}$ is an even polynomial, so $P_{n}$ does not change
its sign at $0$ and $\pi$. Then the number $m$ of zeros from $(-\pi,\pi]$,
where $P_{n}$ changes its sign, is even. Denote the corresponding zeros by
${\beta}_{k},\,1\leqslant k\leqslant m$, where

$\displaystyle-\pi={\beta}_{0}<{\beta}_{1}<\ldots<{\beta}_{m/2}<0<{\beta}_{m/2+1}<\ldots<{\beta}_{m}<{\beta}_{m+1}=\pi.$

Next, by Lemma 6.1, the following equalities are valid for any
$Q_{n}\in\mathcal{T}_{n}$:

$\displaystyle\left\lVert P_{n}\right\rVert_{1}^{*}Q_{n}^{\prime}(0)$
$\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\operatorname{sign}(P_{n}(x))\,Q^{\prime}_{n}(x)dx$
$\displaystyle=$
$\displaystyle\operatorname{sign}(P_{n}(\pi))\sum_{k=0}^{m}(-1)^{k}\int_{\beta_{k}}^{\beta_{k+1}}Q_{n}^{\prime}(x)dx$
$\displaystyle=$
$\displaystyle\operatorname{sign}(P_{n}(\pi))\sum_{k=0}^{m}(-1)^{k}\left(Q_{n}(\beta_{k+1})-Q_{n}(\beta_{k})\right)$
$\displaystyle=$ $\displaystyle
2\operatorname{sign}(P_{n}(\pi))\sum_{k=1}^{m}(-1)^{k+1}Q_{n}(\beta_{k}).$

Furthermore, we show that $m=2n$. Indeed, let us assume that $m/2<n$. Then the
polynomial

$\displaystyle Q_{n,0}(x):=\sin x\prod_{k=1}^{m/2}(\cos x-\cos{\beta}_{k})$

belongs to $\mathcal{T}_{n}$ and
$Q_{n,0}^{\prime}(0)=\prod_{k=1}^{m/2}(1-\cos{\beta}_{k})\neq 0$. On the other
hand, $Q_{n,0}(\beta_{k})=0$ for $1\leqslant k\leqslant m$; therefore,
$Q^{\prime}_{n,0}(0)=0$ by (6). This contradiction shows that $m=2n$, i.e.,
all zeros $\alpha_{k}={\beta}_{k},\,1\leqslant k\leqslant 2n$, of $P_{n}$ are
real and simple. Finally, (6.3) follows from (6). ∎

Note that for $s=0$ and a special class of even trigonometric polynomials,
property (a) was proved in [2, Lemma 6].

##  7\. Appendix: Discrete Hardy Spaces

First, we recall certain facts from [9] about discrete Hardy spaces.

######  Definition 7.1.

(Eoff [9, p. 509]) For a complex-valued sequence
$a=(a_{k})_{k\in\mathbb{Z}}\in\ell_{p}(\mathbb{Z}),\,p\in(0,{\infty})$, let us
define

$\displaystyle H(a)_{m}:=\sum_{k\in\mathbb{Z},k\neq m}\frac{a_{k}}{m-k},\quad
m\in\mathbb{Z};\qquad H(a):=(H(a)_{m})_{m\in\mathbb{Z}}.$

For $p\in(0,{\infty})$, the discrete Hardy space $H_{p}(\mathbb{Z})$ is the
space of all $a\in\ell_{p}(\mathbb{Z})$ for which
$H(a)\in\ell_{p}(\mathbb{Z})$ with the quasi-norm

$\displaystyle\left\lVert a\right\rVert_{H_{p}(\mathbb{Z})}:=\left\lVert
a\right\rVert_{\ell_{p}(\mathbb{Z})}+\left\lVert
H(a)\right\rVert_{\ell_{p}(\mathbb{Z})}.$

In addition, for a complex-valued sequence $a=(a_{k})_{k\in\mathbb{Z}}$, let
us define

(7.1) $f_{a}(x)=\sum_{k\in\mathbb{Z}}(-1)^{k}a_{k}\operatorname{sinc}(x-\pi
k).$

Then the following result holds:

######  Theorem 7.2.

The following statements are valid:

  * (a)

for $p\in(1,{\infty})$, the map $a\mapsto f_{a}$ is an isomorphism between
$\ell_{p}(\mathbb{Z})$ and $E_{p}$;

  * (b)

for $p\in(0,1]$, the map $a\mapsto f_{a}$ is an isomorphism between
$H_{p}(\mathbb{Z})$ and $E_{p}$.

Statement (a) of Theorem 7.2 was proved by Plancherel and Pólya [16] (see also
[9, Theorem 5]), while statement (b) was established by Eoff [9, Theorem 6].

Second, we state a result about the structure of the discrete Hardy spaces.

######  Definition 7.3.

(Boza and Carro [5, Definition 3.9]) Let $p\in(0,1]$. We say that a complex-
valued sequence $a=(a_{k})_{k\in\mathbb{Z}}$ is an $H_{p}$-atom if it
satisfies the following properties:

  1. 1)

$\operatorname{supp}a$ is contained in an interval $I$ with the length
$|I|\geqslant 1$;

  2. 2)

$\left\lVert
a\right\rVert_{\ell_{\infty}(\mathbb{Z})}\leqslant|I|^{-\frac{1}{p}}$;

  3. 3)

$\sum_{k\in\mathbb{Z}}a_{k}k^{j}=0$ for any integer $j$ such that $0\leqslant
j\leqslant p^{-1}-1$.

In particular, it immediately follows from properties 1) and 2) that for an
$H_{p}$-atom $a$,

(7.2) $\left\lVert a\right\rVert_{\ell_{p}(\mathbb{Z})}\lesssim_{p}1,\qquad
p\in(0,1].$

Then the following atomic description for $H_{p}(\mathbb{Z})$ holds:

######  Theorem 7.4.

(Boza and Carro [5, Theorems 3.10 and 3.14]) Let $p\in(0,1]$. Then for any
sequence $a\in H_{p}(\mathbb{Z})$ we have $\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}\asymp_{p}\inf\left\lVert\lambda\right\rVert_{\ell_{p}(\mathbb{Z}_{+})}$,
where
$\lambda=\left(\lambda_{d}\right)_{d\in\mathbb{Z}_{+}}\in\ell_{p}(\mathbb{Z}_{+})$
and the infimum is taken over all possible representations
$a=\sum_{d=0}^{\infty}\lambda_{d}a(d)$ as a linear combination of
$H_{p}$-atoms $\\{a(d)\\}_{d=0}^{\infty}$.

Note that Definitions 7.1, 7.3 and Theorems 7.2, 7.4 are used in Lemma 3.8 and
in the proof of Lemma 3.3.

Third, we define

$\displaystyle H_{c}(a)_{m}:=\sum_{k\in\mathbb{Z}}\frac{a_{k}}{m-k+1/2},\quad
m\in\mathbb{Z};\qquad H_{c}(a):=(H_{c}(a)_{m})_{m\in\mathbb{Z}};$
$\displaystyle\left\lVert a\right\rVert_{H_{p}(\mathbb{Z})}^{*}:=\left\lVert
a\right\rVert_{\ell_{p}(\mathbb{Z})}+\left\lVert
H_{c}(a)\right\rVert_{\ell_{p}(\mathbb{Z})},$

and show that the operator $H_{c}(a)$ is equivalent to $H(a)$ in the
definition of the Hardy space for $p\in(0,1]$. This statement is not trivial
but Eoff [9, p. 509] states this fact without a proof, while using $H_{c}(a)$
in the proof of statement (b) of Theorem 7.2 as well as in the equivalent
definition of $H_{p}(\mathbb{Z})$. That is why we need to prove the following
result.

######  Proposition 7.5.

For any $a\in H_{p}(\mathbb{Z}),\,p\in(0,1]$, the following relation holds:

(7.3) $\left\lVert a\right\rVert_{H_{p}(\mathbb{Z})}^{*}\asymp_{p}\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}.$

###### Proof.

Note first that the proof of the $\gtrsim$ part of the proposition is based on
the following discretization inequalities of Plancherel and Pólya [16, p. 110]
(see also [9, Theorems 2 (ii) and 4 (i)]):

(7.4) $\displaystyle\left(\sum_{n\in\mathbb{Z}}|f(\pi
n)|^{p}\right)^{{1}/{p}}\lesssim_{p}\left\lVert f\right\rVert_{p},\qquad f\in
E_{p},\quad p>0;$ (7.5) $\displaystyle\left(\sum_{n\in\mathbb{Z}}|f(\pi
n)|^{p}\right)^{{1}/{p}}\gtrsim_{p,\tau}\left\lVert
f\right\rVert_{p},\qquad\left(f(\pi
n)\right)_{n\in\mathbb{Z}}\in\ell_{p}(\mathbb{Z}),\quad p>0;$

where $f$ in (7.5) is an entire function of exponential type $\tau\in(0,1)$.

Assume that $\left\lVert a\right\rVert_{H_{p}(\mathbb{Z})}^{*}<{\infty}$. Then
$f_{a}$, defined by (7.1), belongs to $E_{p}$ and

(7.6) $\left\|f_{a}(\cdot/2)\right\|_{p}\lesssim_{p}\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}^{*}.$

Indeed, since $a\in\ell_{2}(\mathbb{Z})$, then
$\left\|f_{a}\right\|_{2}<{\infty}$ and
$\mbox{supp}\,\widehat{f_{a}}\in[-1,1]$. Therefore, $f_{a}(\cdot/2)\in
E_{1/2,2}$. In addition,

$\displaystyle f_{a}(\pi
n/2)=\left\\{\begin{array}[]{ll}(-1)^{m}a_{m},&n=2m,\\\
(-1)^{m}\pi^{-1}H_{c}(a)_{m},&n=2m+1,\end{array}\right.,\qquad
m\in\mathbb{Z}.$

Hence $\left(\sum_{n\in\mathbb{Z}}\left|f_{a}(\pi
n/2)\right|^{p}\right)^{1/p}\lesssim_{p}\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}^{*}$. Thus (7.6) follows from (7.5) for
$\tau=1/2$ and $f=f_{a}(\cdot/2)$ and, in addition, $f_{a}\in E_{p}$ (cf. [9,
Proof of Theorem 6]).

Next, using Bernstein’s inequality (see (1.5) for $s=1$ and $p=q$) and, in
addition, applying inequalities (7.4) and (7.6) to $f_{a}^{\prime}$ and
$f_{a}$, respectively, we have

(7.8) $\left\lVert
H(a)\right\rVert_{\ell_{p}(\mathbb{Z})}=\pi\left(\sum_{m\in\mathbb{Z}}|f^{\prime}_{a}(\pi
m)|^{p}\right)^{{1}/{p}}\lesssim_{p}\left\lVert
f_{a}^{\prime}\right\rVert_{p}\lesssim_{p}\left\lVert
f_{a}\right\rVert_{p}\lesssim_{p}\left\lVert
a\right\rVert^{*}_{H_{p}(\mathbb{Z})}.$

To prove the $\lesssim$ part of the proposition, we first note that the
following estimate for the norm of $H_{c}(a)$ established in [16, Lemme 8, p.
136]:

(7.9) $\left\lVert
H_{c}(a)\right\rVert_{\ell_{p}(\mathbb{Z})}\lesssim_{p}\left\lVert
a\right\rVert_{\ell_{p}(\mathbb{Z})},\qquad p>1,$

is not valid for $p\in(0,1]$. That is why we first estimate $\left\lVert
H(a)\right\rVert_{\ell_{p}(\mathbb{Z})},\,p\in(0,1],$ for an $H_{p}$-atom $a$
(see Definition 7.3) and then discuss any sequence from $H_{p}(\mathbb{Z})$.
Without loss of generality we can assume that
$\mbox{supp}\,a\subseteq[-N,N]\cap\mathbb{Z},\,N\in\mathbb{N}$, with

(7.10) $\displaystyle\left\lVert
a\right\rVert_{\ell_{\infty}(\mathbb{Z})}\lesssim_{p}N^{-1/p},$ (7.11)
$\displaystyle\sum_{k=-N}^{N}a_{k}k^{j}=0,\qquad 0\leqslant j\leqslant
j_{0},\quad j_{0}:=\lfloor 1/p-1\rfloor.$

Then using Hölder’s inequality, estimate (7.9) for $p=2$, and (7.10), we
obtain

(7.12)
$\sum_{m=-2N}^{2N}|H_{c}(a)_{m}|^{p}\lesssim_{p}N^{1-{p}/{2}}\left\lVert
H_{c}(a)\right\rVert_{\ell_{2}(\mathbb{Z})}^{p}\lesssim
N^{1-{p}/{2}}\left\lVert
a\right\rVert_{\ell_{2}(\mathbb{Z})}^{p}\lesssim_{p}1.$

Next, we show that

(7.13) $\sum_{|m|=2N+1}^{\infty}|H_{c}(a)_{m}|^{p}\lesssim_{p}1.$

Indeed, using (7.11), we obtain for $m\geqslant 2N+1$,

$\displaystyle H_{c}(a)_{m}$ $\displaystyle=$
$\displaystyle\sum_{k=-N}^{N}\frac{a_{k}}{m-(k-1/2)}=m^{-1}\sum_{k=-N}^{N}a_{k}\sum_{j=0}^{\infty}\left(\frac{k-1/2}{m}\right)^{j}$
$\displaystyle=$ $\displaystyle
m^{-1}\sum_{j=0}^{\infty}\sum_{k=-N}^{N}a_{k}\left(\frac{k-1/2}{m}\right)^{j}=m^{-1}\sum_{j=j_{0}+1}^{\infty}\sum_{k=-N}^{N}a_{k}\left(\frac{k-1/2}{m}\right)^{j}$
$\displaystyle=$ $\displaystyle
m^{-1}\sum_{k=-N}^{N}a_{k}\sum_{j=j_{0}+1}^{\infty}\left(\frac{k-1/2}{m}\right)^{j}=m^{-j_{0}-1}\sum_{k=-N}^{N}\frac{a_{k}}{m-(k-1/2)}(k-1/2)^{j_{0}+1},$

where the series above are convergent. Hence, taking account of (7.10), we
have

(7.14)
$\left|H_{c}(a)_{m}\right|\lesssim_{p}m^{-j_{0}-2}N^{j_{0}+2-1/p},\qquad
m\geqslant 2N+1,\quad N\in\mathbb{N}.$

It follows from (7.14) that
$\sum_{m=2N+1}^{\infty}|H_{c}(a)_{m}|^{p}\lesssim_{p}1$ since
$p(j_{0}+2)=p(\lfloor 1/p\rfloor+1)>1$. The estimate
$\sum_{m=-2N-1}^{-\infty}|H_{c}(a)_{m}|^{p}\lesssim_{p}1$ can be proved
similarly. Thus, (7.13) is established. Combining together (7.12) with (7.13),
we see that for an $H_{p}$-atom $a$,

(7.15) $\left\lVert H_{c}(a)\right\rVert_{\ell_{p}(\mathbb{Z})}\lesssim_{p}1.$

Furthermore, by Theorem 7.4, any sequence $a\in H_{p}(\mathbb{Z})$ has an
atomic decomposition $a=\sum_{d=0}^{\infty}\lambda_{d}a(d)$, where
$a(0),\,a(1),\ldots,$ are $H_{p}$-atoms, with $\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}\asymp_{p}\left\lVert\lambda\right\rVert_{\ell_{p}(\mathbb{Z}_{+})}$.
In addition, we have by (7.2) that
$\sum_{d=0}^{\infty}\sum_{k\in\mathbb{Z}}|\lambda_{d}||a_{k}(d)|\lesssim_{p}\left\lVert\lambda\right\rVert_{\ell_{1}(\mathbb{Z}_{+})}\leqslant\left\lVert\lambda\right\rVert_{\ell_{p}(\mathbb{Z}_{+})}<\infty$.
Therefore,

(7.16)
$H_{c}\left(\sum_{d=0}^{\infty}\lambda_{d}a(d)\right)_{m}=\sum_{d=0}^{\infty}\lambda_{d}H_{c}\left(a(d)\right)_{m}.$

Thus, using (7.15), (7.16), and Theorem 7.4 we arrive at the relations

(7.17) $\left\lVert
H_{c}(a)\right\rVert^{p}_{\ell_{p}(\mathbb{Z})}\leqslant\inf_{\lambda}\sum_{d=0}^{\infty}\lambda_{d}^{p}\left\lVert
H_{c}(a(d))\right\rVert_{\ell_{p}(\mathbb{Z})}^{p}\lesssim_{p}\inf_{\lambda}\sum_{d=0}^{\infty}\lambda_{d}^{p}\lesssim_{p}\left\lVert
a\right\rVert_{H_{p}(\mathbb{Z})}^{p},$

where the infima in (7.17) are taken over all possible representations
$a=\sum_{d=0}^{\infty}\lambda_{d}a(d)$. Finally, (7.3) follows from (7.8) and
(7.17). ∎

Acknowledgements. We are grateful to Roald M. Trigub for providing reference
[3] and to Kristina Oganesyan for her valuable remarks.

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