Sharp Constants of Approximation Theory. III. Certain Polynomial Inequalities of Different Metrics on Convex Sets

TL;DR

This paper establishes limit relations between sharp polynomial inequalities on convex bodies and entire functions of exponential type, advancing the understanding of approximation theory in multivariate settings.

Contribution

It introduces new limit relations connecting polynomial inequalities on convex bodies with entire functions of exponential type, extending previous univariate results.

Findings

Limit relations between polynomial and entire function inequalities

Sharp constants characterized for multivariate convex bodies

Advancement in approximation theory for multivariate polynomials

Abstract

Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and the corresponding constants for entire functions of exponential type with the spectrum in $V$.

Equations

$$|\beta |:=\sum _{j=1}^m{\beta }_j,|\alpha |:=\sum _{j=1}^m{\alpha }_j,y^{\beta }:=y_1^{{\beta }_1}\cdot \cdot \cdot y_m^{{\beta }_m},D^{\alpha }:=\frac{{\partial }^{{\alpha }_1}}{\partial y_1^{{\alpha }_1}}\cdot \cdot \cdot \frac{{\partial }^{{\alpha }_m}}{\partial y_m^{{\alpha }_m}}.$$

$$\displaystyle\|F\|_{L_{r}(E)}:=\left\{\begin{array}[]{ll}\left(\int_{E}|F(x)|^{r}dx\right)^{1/r},&0<r<{\infty},\\ \mbox{ess}\sup_{x\in E}|F(x)|,&r={\infty}.\end{array}\right.$$

$${\Vert \sum _{j=1}^s{F}_j\Vert }_{L_r(E)}^{\tilde{r}}\le \sum _{j=1}^s{\Vert F_j\Vert }_{L_r(E)}^{\tilde{r}},F_j\in L_r(E),1\le j\le s,$$

$$\Vert y{\Vert }_V^{\ast }:=\underset{t\in V}{\sup }|t\cdot y|,y\in {\mathbb{R}}^m;\Vert z{\Vert }_V^{\ast }:=\underset{t\in V}{\sup }\left|\sum _{j=1}^m{t}_j{z}_j\right|,z\in {\mathbb{C}}^m.$$

$$\Vert y{\Vert }_V^{\ast }=|y|/|v_y|.$$

$$\underset{|z|\to \infty }{\mathrm{limsup}}\log |f(z)|/\sum _{j=1}^m|z_j|<\infty .$$

$$h_f(y,x):=\underset{\rho \to \infty }{\mathrm{limsup}}\log |f(x+i\rho y)|,h_f(y):=\underset{x\in {\mathbb{R}}^m}{\sup }{h}_f(y,x),x\in {\mathbb{R}}^m,y\in {\mathbb{R}}^m,$$

$$M_{p,D_N,n,m,V}:=n^{-N-m/p}\underset{P\in {\mathcal{P}}_{n,m}\setminus \{0\}}{\sup }\frac{|D_N(P)(0)|}{\Vert P{\Vert }_{L_p(V^{\ast })}},$$

$$P_{p,D_N,a,m,V}:=a^{-N-m/p}\underset{T\in {\mathcal{T}}_{aV}\setminus \{0\}}{\sup }\frac{\Vert D_N(T){\Vert }_{L_{\infty }(Q^m(\pi ))}}{\Vert T{\Vert }_{L_p(Q^m(\pi ))}},$$

$$E_{p,D_N,m,V}:=\underset{f\in (B_V\cap L_p({\mathbb{R}}^m))\setminus \{0\}}{\sup }\frac{\Vert D_N(f){\Vert }_{L_{\infty }({\mathbb{R}}^m)}}{\Vert f{\Vert }_{L_p({\mathbb{R}}^m)}}.$$

$$E_{p,D_N,m,V}=\underset{a\to \infty }{\lim }{P}_{p,D_N,a,m,V},p\in (0,\infty ],$$

$$E_{p,D_N,m,V}=\underset{n\to \infty }{\lim }{M}_{p,D_N,n,m,V}.$$

$$\Vert D_N(f_0){\Vert }_{L_{\infty }({\mathbb{R}}^m)}/\Vert f_0{\Vert }_{L_p({\mathbb{R}}^m)}=\underset{n\to \infty }{\lim }{M}_{p,D_N,n,m,V}.$$

$$E_{p,D_N,m,V}=\underset{n\to \infty }{\lim }{M}_{p,D_N,n,m,V}=\underset{a\to \infty }{\lim }{P}_{p,D_N,a,m,V}.$$

$$E_{p,D_N,m,Q^m(1)}=\underset{n\to \infty }{\lim }{n}^{-N-m/p}\underset{P}{\sup }\frac{|D_N(P)(0)|}{\Vert P{\Vert }_{L_p(Q^m(1))}},$$

$$N_{p,D_0,n,m,\Omega }:=n^{-\mu }\underset{P\in {\mathcal{P}}_{n,m}\setminus \{0\}}{\sup }\frac{\Vert P{\Vert }_{L_{\infty }(\Omega )}}{\Vert P{\Vert }_{L_p(\Omega )}},p\in (0,\infty ),$$

$$0<\underset{n\to \infty }{\mathrm{liminf}}{N}_{p,D_0,n,m,\Omega }\le \underset{n\to \infty }{\mathrm{limsup}}{N}_{p,D_0,n,m,\Omega }<\infty ,$$

$${\Vert D^{\alpha }(f)\Vert }_{L_{\infty }({\mathbb{R}}^m)}\le C{\Vert f\Vert }_{L_{\infty }({\mathbb{R}}^m)},f\in B_V\cap L_{\infty }({\mathbb{R}}^m),\alpha \in Z_{+}^m,$$

$${\Vert f\Vert }_{L_{\infty }({\mathbb{R}}^m)}\le C{\Vert f\Vert }_{L_p({\mathbb{R}}^m)},f\in B_V\cap L_p({\mathbb{R}}^m),p\in (0,\infty ),$$

$$|Q_k(z)|\le \frac{C_3(\epsilon ){\left(C_4(1+\epsilon )\Vert z{\Vert }_V^{\ast }\right)}^k}{k!},k\in {\mathbb{Z}}_{+}^1,z\in {\mathbb{C}}^m,$$

$$|Q_k(y)|\le \frac{C_5(\epsilon ){\left((1+\epsilon )\Vert y{\Vert }_V^{\ast }\right)}^k}{k!},k\in {\mathbb{Z}}_{+}^1,y\in {\mathbb{R}}^m,$$

$$\underset{s\to \infty }{\lim }{D}^{\alpha }(f_{n_s})=D^{\alpha }(f_0)$$

$$f_n(z)=\sum _{k=0}^{\infty }{Q}_{k,n}(z)=\sum _{k=0}^{\infty }\sum _{|\beta |=k}{c}_{\beta ,n}{z}^{\beta },n\in \mathbb{N}.$$

$$\underset{s\to \infty }{\lim }{Q}_{k,n_s}=Q_{k,0}$$

$$\underset{z\in d_R}{\max }\left|f_0(z)-f_{n_s}(z)\right|$$

$$\le \underset{z\in d_R}{\max }\sum _{k=0}^{M-1}\left(\left|Q_{k,0}(z)-Q_{k,n_s}(z)\right|\right)+\underset{z\in d_R}{\max }\sum _{k=M}^{\infty }\left(\left|Q_{k,0}(z)|+|Q_{k,n_s}(z)\right|\right)=S_1+S_2,$$

$$S_2\le C_3(1)\sum _{k=M}^{\infty }\frac{(2C_{4}R{)}^k}{k!}.$$

$$|f_0(iy)|\le \sum _{k=0}^{\infty }|Q_{k,0}(y)|\le C_5(\epsilon )\exp ((1+\epsilon )\Vert y{\Vert }_V^{\ast }).$$

$$h_{f_0}(y,x)=\underset{\rho \to \infty }{\mathrm{limsup}}\log |f_0(x+i\rho y)|\le (1+\epsilon )\Vert y{\Vert }_V^{\ast }.$$

$$h_{f_0}(y)=\underset{x\in {\mathbb{R}}^m}{\sup }{h}_{f_0}(y,x)=h_{f_0}(y,x_0)\le (1+\epsilon )\Vert y{\Vert }_V^{\ast }.$$

$$\Vert f-P_k{\Vert }_{L_{\infty }(\tau k{V}^{\ast })}\le C_6(\tau ,m,V)k^{\frac{2m}{m+4}}\exp (-C_7(\tau ,m)k)\Vert f{\Vert }_{L_{\infty }({\mathbb{R}}^m)}.$$

$$\underset{n\to \infty }{\lim }{\Vert D^{\alpha }(f)-D^{\alpha }(P_n)\Vert }_{L_r(\tau n{V}^{\ast })}=0.$$

Full text

1. Introduction

We continue the study of the sharp constants in multivariate inequalities of approximation theory that began in [17] and [18]. In this paper we prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on convex sets and entire functions of exponential type.

Notation. Let ${\mathbb{R}}^{m}$ be the Euclidean $m$-dimensional space with elements $x=(x_{1},\ldots,x_{m}),\,y=(y_{1},\ldots,y_{m}),\,t=(t_{1},\ldots,t_{m}),\,u=(u_{1},\ldots,u_{m})$, the inner product $t\cdot y:=\sum_{j=1}^{m}t_{j}y_{j}$, and the norm $|t|:=\sqrt{t\cdot t}$. Next, ${\mathbb{C}}^{m}:={\mathbb{R}}^{m}+i{\mathbb{R}}^{m}$ is the $m$-dimensional complex space with elements $z=(z_{1},\ldots,z_{m})=x+iy$ and the norm $|z|:=\sqrt{|x|^{2}+|y|^{2}}$; ${\mathbb{Z}}^{m}$ denotes the set of all integral lattice points in ${\mathbb{R}}^{m}$; and ${\mathbb{Z}}^{m}_{+}$ is a subset of ${\mathbb{Z}}^{m}$ of all points with nonnegative coordinates. We also use multi-indices ${\beta}=({\beta}_{1},\ldots,{\beta}_{m})\in{\mathbb{Z}}^{m}_{+}$ and ${\alpha}=({\alpha}_{1},\ldots,{\alpha}_{m})\in{\mathbb{Z}}^{m}_{+}$ with

[TABLE]

Let ${\mathcal{P}}_{n,m}$ be a set of all polynomials $P(y)=\sum_{|{\beta}|\leq n}c_{\beta}y^{\beta}$ in $m$ variables of total degree at most $n,\,n\in Z^{1}_{+}$, with complex coefficients. Given ${\sigma}\in{\mathbb{R}}^{m},\,{\sigma}_{j}>0,\,1\leq j\leq m$, and $M>0$, let $\Pi^{m}({\sigma}):=\{t\in{\mathbb{R}}^{m}:|t_{j}|\leq{\sigma}_{j},1\leq j\leq m\},\,Q^{m}(M):=\{t\in{\mathbb{R}}^{m}:|t_{j}|\leq M,1\leq j\leq m\}$, and ${\mathfrak{B}}^{m}(M):=\{t\in{\mathbb{R}}^{m}:|t|\leq M\}$ be the $m$-dimensional parallelepiped, cube, and ball, respectively.

Let $L_{r}(E)$ be the space of all measurable complex-valued functions $F$ on a measurable set $E\subseteq{\mathbb{R}}^{m}$ with the finite quasinorm

[TABLE]

This quasinorm allows the following ”triangle” inequality:

[TABLE]

where $s\in{\mathbb{N}}:=\{1,\,2,\ldots\}$ and $\tilde{r}:=\min\{1,r\}$ for $r\in(0,{\infty}]$.

In this paper we will need certain definitions and properties of convex bodies in ${\mathbb{R}}^{m}$. We first define the width $w(K)$ of a convex body $K$ in ${\mathbb{R}}^{m}$ as the minimal distance between two parallel supporting hyperplanes of $K$.

Next, throughout the paper $V$ is a centrally symmetric (with respect to the origin) closed convex body in ${\mathbb{R}}^{m}$ and $V^{*}:=\{y\in{\mathbb{R}}^{m}:\forall\,t\in V,|t\cdot y|\leq 1\}$ is the polar of $V$. It is well known that $V^{*}$ is a centrally symmetric (with respect to the origin) closed convex body in ${\mathbb{R}}^{m}$ and $V^{**}=V$ (see, e.g., [29, Sect. 14]). The set $V$ generates the following dual norms on ${\mathbb{R}}^{m}$ and ${\mathbb{C}}^{m}$ by

[TABLE]

Note also that $V^{*}$ is the unit ball in the norm $\|\cdot\|_{V}^{*}$ on ${\mathbb{R}}^{m}$ with the boundary $\partial(V^{*})$, the width $w(V^{*})$, and the $m$-dimensional volume $|V^{*}|_{m}$. For example, if ${\sigma}\in{\mathbb{R}}^{m},\,{\sigma}_{j}>0,\,1\leq j\leq m$, and $V=\left\{t\in{\mathbb{R}}^{m}:\left(\sum_{j=1}^{m}|t_{j}/{\sigma}_{j}|^{\mu}\right)^{1/\mu}\leq 1\right\}$, then for $y\in{\mathbb{R}}^{m},\,\|y\|_{V}^{*}=\left(\sum_{j=1}^{m}|{\sigma}_{j}y_{j}|^{\lambda}\right)^{1/{\lambda}}$ and $V^{*}=\left\{y\in{\mathbb{R}}^{m}:\left(\sum_{j=1}^{m}|{\sigma}_{j}y_{j}|^{\lambda}\right)^{1/{\lambda}}\leq 1\right\}$, where $\mu\in[1,{\infty}],\,{\lambda}\in[1,{\infty}]$, and $1/\mu+1/{\lambda}=1$. In particular, $\|y\|_{\Pi^{m}({\sigma})}^{*}=\sum_{j=1}^{m}{\sigma}_{j}|y_{j}|,\,\|y\|_{Q^{m}(M)}^{*}=M\sum_{j=1}^{m}|y_{j}|$, and $\|y\|_{{\mathfrak{B}}^{m}(M)}^{*}=M|y|$.

Let $l_{y}$ be a straight line passing through a fixed point $y\in{\mathbb{R}}^{m}$ and the origin, and let $v_{y}\in l_{y}\cap\partial(V^{*})$ be the closest point to $y$. We also need the equivalent definition of the dual norm in ${\mathbb{R}}^{m}$ given by the formula (see, e.g., [29, Theorem 14.5])

[TABLE]

Given $a>0$, the set of all trigonometric polynomials $T(x)=\sum_{k\in aV\cap{\mathbb{Z}}^{m}}c_{k}e^{ik\cdot x}$ with complex coefficients is denoted by ${\mathcal{T}}_{aV}$.

Definition 1.1**.**

We say that an entire function $f:{\mathbb{C}}^{m}\to{\mathbb{C}}^{1}$ has exponential type $V$ if for any ${\varepsilon}>0$ there exists a constant $C_{0}({\varepsilon},f)$ such that for all $z\in{\mathbb{C}}$, $|f(z)|\leq C_{0}({\varepsilon},f)\exp\left((1+{\varepsilon})\|z\|_{V}^{*}\right)$.

The class of all entire function of exponential type $V$ is denoted by $B_{V}$. It is easy to verify that if $V_{1}\subseteq V_{2}$, then $B_{V_{1}}\subseteq B_{V_{2}}$. Throughout the paper, if no confusion may occur, the same notation is applied to $f\in B_{V}$ and its restriction to ${\mathbb{R}}^{m}$ (e.g., in the form $f\in B_{V}\cap L_{p}({\mathbb{R}}^{m}))$. The class $B_{V}$ was defined by Stein and Weiss [33, Sect. 3.4]. For $V=\Pi^{m}({\sigma}),\,V=Q^{m}(M),$ and $V={\mathfrak{B}}^{m}(M)$, similar classes were defined by Bernstein [5] and Nikolskii [26], [27, Sects. 3.1, 3.2.6], see also [9, Definition 5.1]. Properties of functions from $B_{V}$ have been investigated in numerous publications (see, e.g., [5, 26, 27, 33, 25, 12, 13, 14] and references therein). Some of these properties are presented in Section 2. We need more definitions for entire functions of several variables.

We say that an entire function $f:{\mathbb{C}}^{m}\to{\mathbb{C}}^{1}$ has exponential type if

[TABLE]

The class of all entire functions of exponential type is denoted by $B$.

Next, let us define the indicator and the $P$-indicator of $f\in B$ by

[TABLE]

respectively. Note that equivalent definitions of $h_{f}(y,x)$ and $h_{f}(y)$ were introduced by Plancherel and Pólya [28] (see also [31, Sect. 3.4.2]). The class of all functions $f\in B$ with $h_{f}(y)\leq\|y\|_{V}^{*},\,y\in{\mathbb{R}}^{m},$ is denoted by $B^{*}_{V}$. A similar class was introduced by the author [12]. It turns out that $B_{V}^{*}\cap L_{p}({\mathbb{R}}^{m})=B_{V}\cap L_{p}({\mathbb{R}}^{m}),\,p\in(0,{\infty}]$ (see Lemma 2.1 (d)), but the definition of $B_{V}^{*}$ is needed for the proof of important Lemma 2.2.

Throughout the paper $C,\,C_{0},\,C_{1},\ldots$ denote positive constants independent of essential parameters. Occasionally we indicate dependence on certain parameters. The same symbol $C$ does not necessarily denote the same constant in different occurrences. In addition, we use the ceiling function $\lceil a\rceil$.

** Markov-Bernstein-Nikolskii Type Inequalities.** Let $D_{N}:=\sum_{|{\alpha}|=N}b_{\alpha}D^{\alpha}$ be a linear differential operator with constant coefficients $b_{\alpha}\in{\mathbb{C}}^{1},\,|{\alpha}|=N,\,N\in{\mathbb{Z}}^{1}_{+}$. We assume that $D_{0}$ is the identity operator.

Next, we define sharp constants in multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic and trigonometric polynomials and entire functions of exponential type. Let

[TABLE]

Here, $a>0,\,N\in{\mathbb{Z}}^{1}_{+},\,V\subset{\mathbb{R}}^{m}$, and $p\in(0,{\infty}]$.

The purpose of this paper is to prove limit relations between $E_{p,D_{N},m,V}$ and $M_{p,D_{N},n,m,V}$. The limit relation for multivariate trigonometric polynomials

[TABLE]

was proved by the author [17, Theorem 1.3]. In the univariate case of $V=[-1,1],\,D_{N}=d^{N}/dx^{N}$, and $a\in{\mathbb{N}}$, (1.6) was proved by the author and Tikhonov [19]. In earlier publications [21, 22], Levin and Lubinsky established versions of (1.6) on the unit circle for $N=0$. Certain extensions of the Levin-Lubinsky’s results to the $m$-dimensional unit sphere in ${\mathbb{R}}^{m+1}$ were recently proved by Dai, Gorbachev, and Tikhonov [8].

In case of an even $N\in{\mathbb{Z}}^{1}_{+},\,p\in[1,{\infty}]$, the unit ball $V={\mathfrak{B}}^{m}(1)$, and the operator $D_{N}=\Delta^{N/2}$, where $\Delta$ is the Laplace operator, the author [18, Corollary 4.4] proved the following limit relation for multivariate algebraic polynomials:

[TABLE]

The proof of (1.7) was based on invariance theorems and limit relations for sharp constants in univariate weighted spaces (see [18, Theorems 2.1, 2.2, 4.1]). Note that certain properties of the sharp constants in univariate weighted spaces are discussed by Arestov and Deikalova [2]. For $m=1,\,V=[-1,1],\,D_{N}=d^{N}/dx^{N},\,N\in{\mathbb{Z}}^{1}_{+}$, and $p\in(0,{\infty}]$, (1.7) was established in [16, Theorem 1.1].

In this paper we extend relation (1.7) to any $N\in{\mathbb{Z}}^{1}_{+},\,p\in(0,{\infty}],\,V\subset{\mathbb{R}}^{m}$, and $D_{N}$.

Main Results and Remarks. We recall that $V$ is a centrally symmetric (with respect to the origin) closed convex body in ${\mathbb{R}}^{m}$.

Theorem 1.2**.**

If $n\in{\mathbb{N}},\,N\in{\mathbb{Z}}^{1}_{+},\,V\subset{\mathbb{R}}^{m}$, and $p\in(0,{\infty}]$, then $\lim_{n\to{\infty}}M_{p,D_{N},n,m,V}$ exists and

[TABLE]

In addition, there exists a nontrivial function $f_{0}\in B_{V}\cap L_{p}({\mathbb{R}}^{m})$ such that

[TABLE]

The following corollary is a direct consequence of Theorem 1.2 and relation (1.6).

Corollary 1.3**.**

If $n\in{\mathbb{N}},\,N\in{\mathbb{Z}}^{1}_{+},\,V\subset{\mathbb{R}}^{m}$, and $p\in(0,{\infty}]$, then

[TABLE]

Remark 1.4*.*

Relations (1.8) and (1.9) show that the function $f_{0}\in B_{V}\cap L_{p}({\mathbb{R}}^{m})$ from Theorem 1.2 is an extremal function for $E_{p,D_{N},m,V}$.

Remark 1.5*.*

In definitions (1.4) and (1.5) of the sharp constants we discuss only complex-valued functions $P$ and $f$. We can define similarly the ”real” sharp constants if the suprema in (1.4) and (1.5) are taken over all real-valued functions on ${\mathbb{R}}^{m}$ from ${\mathcal{P}}_{n,m}\setminus\{0\}$ and $(B_{V}\cap L_{p}({\mathbb{R}}^{m}))\setminus\{0\}$, respectively. It turns out that the ”complex” and ”real” sharp constants coincide. For $m=1$ this fact was proved in [16, Sect. 1] (cf. [19, Theorem 1.1]) and the case of $m>1$ can be proved similarly.

Remark 1.6*.*

Note that the following relation for a different class of polynomials holds true:

[TABLE]

where the supremum in (1.10) is taken over all nontrivial polynomials $P$ in $m$ variables of degree at most $n$ in each variable. In addition, note that, unlike relation (1.8), both sides of (1.10) contain the same set $V=Q^{m}(1)$.

Remark 1.7*.*

Equality (1.8) presents one more example of a relation between entire functions of exponential type $V$ on ${\mathbb{R}}^{m}$ and polynomials on $V^{*}$. The limit relations for the corresponding errors of approximation of $f\in L_{p}({\mathbb{R}}^{m})$ by functions from $B_{V}$ and polynomials from ${\mathcal{P}}_{n,m}$ restricted to $nV^{*}$ were proved by the author [12, Theorems 5.1, 5.2]. Certain versions of these results are discussed below in Lemmas 2.3 and 2.4.

Remark 1.8*.*

Let us introduce a sharp constant in the classic inequality of different metrics

[TABLE]

where $\Omega$ is a compact subset of ${\mathbb{R}}^{m}$ and $\mu=\mu(\Omega,p,m)>0$ is a constant. Let us assume that the following inequalities hold true:

[TABLE]

if any. For example, the third inequality in (1.11) holds true for $\mu=2m/p$ and any domain $\Omega$, satisfying the cone condition, in particular for convex bodies (not necessarily symmetric), see [6, Theorem 1], [7, Theorem 2], [11, Theorem 2]. It is also valid for $\mu=(m+1)/p$ and any domain $\Omega$ with the smooth boundary, see [6, Theorem 2], [7, Theorem 5], [20, p. 433]. The typical examples of sets $\Omega$ when all inequalities in (1.11) hold true are the unit cube $\Omega=Q^{m}(1)$ for $\mu=2m/p$ and the unit ball $\Omega={\mathfrak{B}}^{m}(1)$ for $\mu=(m+1)/p$. More examples and discussions are presented by Ditzian and Prymak [10]. The author (see [16, Theorem 1.4] and [18, Corollary 4.6]) proved that $\lim_{n\to{\infty}}N_{p,D_{0},n,1,[-1,1]}$ exists for $\mu=2/p$ and found its exact value. However, it is unknown as to whether $\lim_{n\to{\infty}}N_{p,D_{0},n,m,\Omega}$ exists for $m>1$.

The proof of Theorem 1.2 is presented in Section 3. Section 2 contains certain properties of functions from $B_{V}$ and ${\mathcal{P}}_{n,m}$. In particular, Lemma 2.7 that discusses inequalities of different metrics for multivariate polynomials on smaller domains is of independent interest.

2. Properties of Entire Functions and

Polynomials

In this section we discuss certain properties of multivariate entire functions of exponential type and polynomials that are needed for the proof of Theorem 1.2. We start with certain properties of entire functions of exponential type.

Lemma 2.1**.**

*(a) If $f\in B_{V}$, then there exists $M=M(V)$ such that $f\in B_{Q^{m}(M)}$.

(b) $f\in B$ if and only if there exists $V\subset{\mathbb{R}}^{m}$ such that $f\in B_{V}$.

(c) The following Bernstein and Nikolskii type inequalities hold true:*

[TABLE]

*where $C$ is independent of $f$.

(d) For $p\in(0,{\infty}],\,B_{V}\cap L_{p}({\mathbb{R}}^{m})=B_{V}^{*}\cap L_{p}({\mathbb{R}}^{m})$.

(e) The Lebesgue measure of the set $\{x\in{\mathbb{R}}^{m}:h_{f}(y,x)<h_{f}(y)\}$ is zero for every $y\in{\mathbb{R}}^{m}$.*

Proof.

Statement (a) follows from the obvious inclusion $V\subseteq Q^{m}(M)$ for a certain $M=M(V)>0$. Next, statement (b) follows from the inequalities $C_{1}\sum_{j=1}^{m}|z_{j}|\leq\|z\|_{V}^{*}\leq C_{2}\sum_{j=1}^{m}|z_{j}|,$ where $C_{1}=C_{1}(V):=1/(2\max\{1,{\lambda}\})$ (here, ${\lambda}$ is the least number such that $Q^{m}(1)\subseteq{\lambda}V$) and $C_{2}=C_{2}(V):=\max_{1\leq j\leq m}\max_{t\in V}|t_{j}|$. Inequality (2.1) for $V=Q^{m}(M),\,M>0$, is well known (see, e.g., [27, Eq. 3.2.2(8)]). Then (2.1) follows from statement (a). Inequality (2.2) was proved in [25, Theorem 5.7].

To prove statement (d), we note that if $f\in B_{V}\cap L_{p}({\mathbb{R}}^{m}),\,p\in(0,{\infty})$, then $f\in B_{V}\cap L_{\infty}({\mathbb{R}}^{m})$ by (2.2), while the relation $B_{V}\cap L_{\infty}({\mathbb{R}}^{m})=B_{V}^{*}\cap L_{\infty}({\mathbb{R}}^{m})$ was proved in a more general form by the author [12, Corollary 2.1]. Statement (e) was proved by Ronkin (see [30] and [31, Theorem 3.4.3]). $\Box$

A compactness theorem for a set of entire functions is discussed below.

Lemma 2.2**.**

(a) Let ${\mathcal{E}}_{m}$ be the set of all multivariate entire functions $f(z)=\sum_{k=0}^{\infty}Q_{k}(z)$, where $Q_{k}(z)=\sum_{|{\beta}|=k}c_{\beta}z^{\beta}$ is the homogeneous polynomial of degree $k\in{\mathbb{Z}}_{+}^{1}$, satisfying the following conditions: for any ${\varepsilon}>0$ the following inequalities are valid:

[TABLE]

where $C_{3}({\varepsilon})$ and $C_{5}({\varepsilon})$ are independent of $z,\,f$ and $k$, and $C_{4}$ is independent of $z,\,{\varepsilon},\,f$ and $k$. Then for any sequence $\{f_{n}\}_{n=1}^{\infty}\subseteq{\mathcal{E}}_{m}$ there exist a subsequence $\{f_{n_{s}}\}_{s=1}^{\infty}$ and a function $f_{0}\in B_{V}^{*}$ such that for every ${\alpha}\in{\mathbb{Z}}_{+}^{m}$,

[TABLE]

*uniformly on each compact subset of ${\mathbb{C}}^{m}$.

(b) If the function $f_{0}$ from statement (a) belongs to $L_{\infty}({\mathbb{R}}^{m})$, then $f_{0}\in B_{V}\cap L_{\infty}({\mathbb{R}}^{m})$.*

Proof.

(a) By a multivariate version of Weierstrass’ theorem (see, e.g., [32, Sect. 1.2.5]), it suffices to prove statement (a) for ${\alpha}=0$. Let us set $d_{R}:=\{z\in{\mathbb{C}}^{m}:\|z\|_{V}^{*}\leq R\},\,R>0$, and let

[TABLE]

By (2.3), the polynomials $Q_{k,n},\,n\in{\mathbb{N}}$, are uniformly bounded on $d_{R}$ for a fixed $R\geq 0$ and each $k\in{\mathbb{Z}}^{1}_{+}$. Therefore, using the Cantor diagonal process, we can select a subsequence $\{f_{n_{s}}\}_{s=1}^{\infty}$ such that

[TABLE]

uniformly on $d_{R}$ for any $R\geq 0$ and $k\in{\mathbb{Z}}^{1}_{+}$. Here, $Q_{k,0}=\sum_{|{\beta}|=k}c_{{\beta},0}z^{\beta}$ are homogeneous polynomials of degree $k,\,k\in{\mathbb{Z}}^{1}_{+}$. Then inequality (2.3) shows that $f_{0}(z)=\sum_{k=0}^{\infty}Q_{k,0}(z)$ is an entire function that belongs to the class $B_{C_{4}V}$. Further, we obtain for $M\in{\mathbb{N}}$

[TABLE]

where by (2.3) for ${\varepsilon}=1$ and by (2.6),

[TABLE]

Then given ${\delta}>0$ and $R>0$, we can choose $M=M({\delta},R)$ such that $S_{2}<{\delta}/2$. Finally by (2.6), we can choose $s_{0}=s_{0}({\delta},R)\in{\mathbb{N}}$ such that $S_{1}<{\delta}/2$ for all $s\geq s_{0}$. Thus (2.5) holds uniformly on $d_{R}$, where $f_{0}\in B_{C_{4}V}$. In addition, $f_{0}\in B$ by Lemma 2.1 (b).

Next, we prove that $f_{0}\in B^{*}_{V}$. It follows from (2.4) and (2.6) that for any ${\varepsilon}>0$ and $y\in{\mathbb{R}}^{m}$,

[TABLE]

The function $f_{0}(z)\exp(-(1+{\varepsilon})\|z\|^{*}_{V})$ is continuous at $z=iy,\,y\in{\mathbb{R}}^{m}$. Hence by (2.7), there exists a number ${\varepsilon}_{1}={\varepsilon}_{1}(y)$ such that for any $x\in R^{m}({\varepsilon}_{1}),\,|f_{0}(x+iy)|\leq 2C_{5}({\varepsilon})\exp((1+{\varepsilon})\|y\|^{*}_{V})$. Therefore, for a fixed $y\in{\mathbb{R}}^{m}$ and any $x\in R^{m}({\varepsilon}_{1}),$

[TABLE]

By Lemma 2.1 (e), the Lebesgue measure of the set $\{x\in{\mathbb{R}}^{m}:h_{f_{0}}(y,x)<h_{f_{0}}(y)\}$ is equal to zero for every fixed $y\in{\mathbb{R}}^{m}$. Hence using (2.8), we see that there exists $x_{0}=x_{0}(y)\in R^{m}({\varepsilon}_{1})$ such that

[TABLE]

Thus $f_{0}\in\bigcap_{{\delta}>0}B_{(1+{\varepsilon})V}^{*}=B_{V}^{*}$. This completes the proof of statement (a) of the lemma.

(b) If $f_{0}\in L_{\infty}({\mathbb{R}}^{m})$, then $f_{0}\in B_{V}^{*}\cap L_{\infty}({\mathbb{R}}^{m})$ by statement (a), and $f_{0}\in B_{V}\cap L_{\infty}({\mathbb{R}}^{m})$ by Lemma 2.1 (d). $\Box$

In the next two lemmas we discuss estimates of the error of polynomial approximation for functions from $B_{V}$.

Lemma 2.3**.**

For any function $f\in B_{V}\cap L_{\infty}({\mathbb{R}}^{m}),\,\tau\in(0,1)$, and $k\in{\mathbb{N}}$, there is a polynomial $P_{k}\in{\mathcal{P}}_{k,m}$ such that

[TABLE]

This result was proved by the author [12, Lemma 4.4]. In case of $m=1$ a version of Lemma 2.3 was established by Bernstein [4] (see also [34, Sect. 5.4.4] and [1, Appendix, Sect. 83]). More general and more precise inequalities were obtained in [12] and [13].

Lemma 2.4**.**

For any $f\in B_{V}\cap L_{\infty}({\mathbb{R}}^{m}),\,\tau\in(0,1)$, and $n\in{\mathbb{N}}$, there is a polynomial $P_{n}\in{\mathcal{P}}_{n,m}$ such that for ${\alpha}\in{\mathbb{Z}}_{+}^{m}$ and $r\in(0,{\infty}]$,

[TABLE]

Proof.

First of all, for $P_{k}\in{\mathcal{P}}_{k,m},\,k\in Z_{+}^{1},\,a>0$, and ${\alpha}\in{\mathbb{Z}}_{+}^{m}$, we need the following crude Markov-type inequality:

[TABLE]

Inequality (2.11) easily follows from a multivariate A. A. Markov-type inequality proved by Wilhelmsen [35, Theorem 3.1].

Next, let $\{P_{k}\}_{k=1}^{\infty}$ be the sequence of polynomials from Lemma 2.3. Then using (2.11) for $a=\tau k$ and (2.9), we obtain

[TABLE]

Hence for $n\in{\mathbb{N}},\,{\alpha}\in{\mathbb{Z}}_{+}^{m}$, and $r\in(0,{\infty}]$ we have

[TABLE]

Thus (2.10) is established. $\Box$

Certain inequalities for multivariate trigonometric and algebraic polynomials are discussed in the next three lemmas.

Lemma 2.5**.**

For a polynomial $P(y)=\sum_{|{\beta}|\leq n}c_{\beta}y^{\beta}\in{\mathcal{P}}_{n,m},\,n\in{\mathbb{Z}}^{1}_{+}$, the following inequalities hold true:

[TABLE]

where $C_{4}$ is independent of $z,\,a,\,f$, and $k$.

Proof.

Inequality (2.12) is proved in [12, Lemma 3.2]. We first prove inequality (2.13) for $m=1$ and $V=[-1,1]$, i.e., we prove the inequality

[TABLE]

which is equivalent to

[TABLE]

where $M:=\max_{\tau\in[-a,a]}\left|\sum_{k=0}^{n}d_{k}\tau^{k}\right|$. This inequality without proof was discussed in [4] and [34, Eq. 2.6(9)] as a corollary of V. A. Markov’s inequality [23] (see also [34, Sect. 2.9.12]) and [24, Eqs. (5.1.4.1) and (6.1.2.5)])

[TABLE]

where $T_{p}\in{\mathcal{P}}_{p,1}$ is the Chebyshev polynomial of the first kind. It is sufficient to derive (2.15) from (2.16) for $M=1$ and $a=1$. We consider three cases.

Case 1: $k=2p,\,n=2N$ or $n=2N+1,\,0\leq p\leq N$. Then we have from (2.16)

[TABLE]

Case 2: $k=2p+1,\,n=2N+1,\,0\leq p\leq N$. Then we have from (2.16)

[TABLE]

Case 3: $k=2p+1,\,n=2N,\,0\leq p\leq N-1$. It follows from (2.16) that we can replace $N$ with $N-1$ in inequalities (2.22). Therefore,

[TABLE]

Thus (2.15) and (2.14) follow from (2.19), (2.22), and (2.24).

To prove (2.13) for $m>1$, we first draw a line $l_{y}$ passing through a fixed point $y\in{\mathbb{R}}^{m}$ and the origin with the equation $t=\tau v_{y}$, where $\tau\in{\mathbb{R}}^{1}$ and $v_{y}\in l_{y}\cap\partial(aV^{*})$ is the closest point to $y$. In particular, $|\tau|\leq 1$ for $t\in l_{y}\cap aV^{*}$. In addition, we see by (1.3) that for $t=y$,

[TABLE]

Next, the restriction of $P\in{\mathcal{P}}_{n,m}$ to $l_{y}$ is a polynomial $p(\tau)=\sum_{k=0}^{n}d_{k}\tau^{k}\in{\mathcal{P}}_{n,1}$, where $d_{k}=\sum_{|{\beta}|=k}c_{\beta}v_{y}^{\beta},\,0\leq k\leq n$, and using relations (2.14) and (2.25), we obtain

[TABLE]

Thus (2.13) is established. $\Box$

Note that a version of Lemma 2.5 was proved in [15, Theorem 1].

Next, we need a multivariate version of a generalized Bari’s inequality [16, Lemma 2.4]. Bari [3, Theorem 6] proved the following result for $m=1$ and $p\in[1,{\infty})$.

Lemma 2.6**.**

Let $T\in{\mathcal{T}}_{nQ^{m}(1)}$ be an even trigonometric polynomial of degree at most $n$ in each variable, $n\in{\mathbb{N}}$, and let $0\leq a_{j}<a_{1,j}\leq b_{1,j}<b_{j}\leq\pi,\,1\leq j\leq m$. Then for $p\in(0,{\infty})$,

[TABLE]

where $C_{10}$ is independent of $n$ and $T$.

Proof.

The proof follows that of [3, Theorem 6] and [16, Lemma 2.4]. Let $t_{j}=t_{j}(u_{j}):[a_{j},b_{j}]\to[0,\pi]$ be the one-to-one function defined by the equation

[TABLE]

and let ${\lambda}_{1,j}=t_{j}(a_{1,j}),\,{\lambda}_{2,j}=t_{j}(b_{1,j})$, where $0<{\lambda}_{1,j}\leq{\lambda}_{2,j}<\pi,\,1\leq j\leq m$. Then

[TABLE]

and

[TABLE]

Next, let $H_{n}(t):=T(u)$, by substitution (2.27), and let $G_{n+d}(t):=H_{n}(t)\prod_{j=1}^{m}\sin^{d}t_{j}\in{\mathcal{T}}_{(n+d)Q^{m}(1)}$, where $d:=\lceil 1/p\rceil$, i.e., $dp\geq 1$. Then using (2.28), (2.29), and Nikolskii’s inequality for multivariate trigonometric polynomials of degree at most $n+d$ in each variable [27, Sect. 3.3.3], we obtain

[TABLE]

Hence (2.26) follows. $\Box$

Certain polynomial estimates based on Lemma 2.6 are discussed in the following lemma.

Lemma 2.7**.**

*Let $P$ be an algebraic polynomial in $m$ variables of degree at most $n$ in each variable.

(a) For $p\in(0,{\infty})$ and $M>0$,*

[TABLE]

(b) For $p\in(0,{\infty}),\,a>0$, and ${\varepsilon}>0$, the following inequality holds true:

[TABLE]

Proof.

Setting

[TABLE]

we have by Lemma 2.6

[TABLE]

Hence (2.30) holds true.

(b) Let $y_{0}\in aV^{*}$ such that $\|P\|_{L_{\infty}(aV^{*})}=\left|P(y_{0})\right|$. The cube $Q_{M}:=Q^{m}(aM)+y_{0}$, where $M:={\varepsilon}\,a\,w(V^{*})/(2\sqrt{m})$, is a subset of $(1+{\varepsilon})aV^{*}$. Indeed, for any $y\in Q_{M}$ we have by (1.3),

[TABLE]

Therefore, it follows from (2.30) that

[TABLE]

This proves (2.31). $\Box$

Remark 2.8*.*

Lemma 2.7 (a) for $p\in[1,{\infty})$ was proved by a different method in [12, Lemma 3.3]. Note that (2.31) is valid for any convex body $V^{*}$. In addition, note that an estimate of the constant in Nikolskii-type inequality (2.31) on a smaller convex body is $C(n/a)^{m/p}$, while for the corresponding inequality on the same domain the estimate is $C(n/a)^{(m+1)/p}$ for domains with smooth boundaries and the estimate is $C(n/a)^{2m/p}$ for domains with boundaries, satisfying the cone condition (see Remark 1.8).

3. Proof of Theorem 1.2

Throughout the section we use the notation $\tilde{p}=\min\{1,p\},\,p\in(0,{\infty}]$, introduced in Section 1.

Proof of Theorem 1.2. We first prove the inequality

[TABLE]

Let $f$ be any function from $B_{V}\cap L_{p}({\mathbb{R}}^{m}),\,p\in(0,{\infty}]$. Then $f\in B_{Q^{m}(M)}$ by Lemma 2.1 (a); hence $D_{N}(f)\in B_{Q^{m}(M)}$ by [27, Sect. 3.1] (see also [17, Lemma 2.1 (d)]). In addition, $D_{N}(f)\in L_{p}({\mathbb{R}}^{m})$ by Bernstein’s and Nikolskii’s inequalities (2.1) and (2.2) and by the ”triangle” inequality (1.2). Therefore,

[TABLE]

Indeed, since $D_{N}(f)\in B_{Q^{m}(M)}\cap L_{p}({\mathbb{R}}^{m})$, (3.2) is known for $p\in[1,{\infty})$ (see, e.g., [27, Theorem 3.2.5]), and for $p\in(0,1)$ it follows from Nikolskii’s inequality (2.2), since if $D_{N}(f)\in L_{p}({\mathbb{R}}^{m}),\,p\in(0,1)$, then $D_{N}(f)\in L_{1}({\mathbb{R}}^{m})$.

Let us first prove (3.1) for $p\in(0,{\infty})$. Then by (3.2), there exists $x_{0}\in{\mathbb{R}}^{m}$ such that $\|D_{N}(f)\|_{L_{\infty}({\mathbb{R}}^{m})}=\left|D_{N}(f)(x_{0})\right|$. Without loss of generality we can assume that $x_{0}=0$. Let $\tau\in(0,1)$ be a fixed number. Then using polynomials $P_{n}\in{\mathcal{P}}_{n,m},\,n\in{\mathbb{N}}$, from Lemma 2.4, we obtain for $r={\infty}$ by (1.4),

[TABLE]

Next, note that $f\in L_{\infty}({\mathbb{R}}^{m})$, by Nikolskii’s inequality (2.3). Using again Lemma 2.4 (for ${\alpha}=0$ and $r=p$), we have from (1.2)

[TABLE]

Combining (3) with (3.4), and letting $\tau\to 1-$, we arrive at (3.1) for $p\in(0,{\infty})$.

In the case $p={\infty}$, for any ${\varepsilon}>0$ there exists $x_{0}\in{\mathbb{R}}^{m}$ such that $\|D_{N}(f)\|_{L_{\infty}({\mathbb{R}}^{m})}<(1+{\varepsilon})\left|D_{N}(f)(x_{0})\right|$. Without loss of generality we can assume that $x_{0}=0$. Then similarly to (3) and (3.4) we can obtain the inequality

[TABLE]

Finally letting $\tau\to 1-$ and ${\varepsilon}\to 0+$ in (3.5), we arrive at (3.1) for $p={\infty}$. This completes the proof of (3.1).

Further, we will prove the inequality

[TABLE]

by constructing a nontrivial function $f_{0}\in B_{V}\cap L_{p}({\mathbb{R}}^{m})$, such that

[TABLE]

Then inequalities (3.1) and (3.6) imply (1.8). In addition, $f_{0}$ is an extremal function in (1.8), that is, (1.9) is valid.

It remains to construct a nontrivial function $f_{0}$, satisfying (3.7). We first note that

[TABLE]

This inequality follows immediately from (3.1). Let $P_{n}\in{\mathcal{P}}_{n,m}$ be a polynomial, satisfying the equality

[TABLE]

The existence of an extremal polynomial $P_{n}$ in (3.9) can be proved by the standard compactness argument (cf. [19]). Next, setting $Q_{n}(y):=P_{n}(y/n)=\sum_{|{\beta}|\leq n}c_{\beta}y^{\beta}$, we have from (3.9) that

[TABLE]

We can assume that

[TABLE]

Then it follows from (3.10), (3.11), and (3.8) that

[TABLE]

Further, for any ${\varepsilon}>0$ and $0\leq k\leq n$, the following inequalities hold true:

[TABLE]

Inequalities (3.13) and (3.14) follow from (3.12) and Lemmas 2.5 and 2.7 (b) for $a=n/(1+{\varepsilon})$. Let $\{n_{r}\}_{r=1}^{\infty}$ be a subsequence of ${\mathbb{N}}$ such that

[TABLE]

Inequalities (3.13) and (3.14) show that the polynomial sequence $\{Q_{n_{r}}\}_{r=1}^{\infty}\subseteq{\mathcal{E}}_{m}$ satisfies the conditions of Lemma 2.2. Therefore, there exist a subsequence $\{Q_{n_{r_{s}}}\}_{s=1}^{\infty}$ and a function $f_{0}\in B_{V}$ such that

[TABLE]

uniformly on any cube $Q^{m}(M),\,M>0$. Moreover, by (3.11) and (3.16),

[TABLE]

In addition, using (1.2), (3.16), (3.10), (3.11), and (3.15), we obtain for any cube $Q^{m}(M),\,M>0$,

[TABLE]

Next using (3) and (3.8), we see that

[TABLE]

Therefore, $f_{0}$ is a nontrivial function from $B_{V}\cap L_{p}({\mathbb{R}}^{m})$, by (3.17) and (3.19). Thus for any cube $Q^{m}(M),\,M>0$, we obtain from (3.15), (3.16), and (3.17)

[TABLE]

Finally, letting $M\to{\infty}$ in (3.20), we arrive at (3.7). $\Box$

Acknowledgements. We are grateful to the anonymous referee for valuable suggestions.