Sharp Constants of Approximation Theory. II. Invariance Theorems and Certain Multivariate Inequalities of Different Metrics

TL;DR

This paper establishes invariance theorems for multivariate inequalities involving sharp constants, connecting polynomial inequalities on spheres and balls with entire functions, advancing approximation theory in multiple metrics.

Contribution

It introduces invariance theorems that relate sharp constants in multivariate polynomial inequalities to entire functions, expanding understanding of approximation bounds across different metrics.

Findings

Proved invariance theorems for inequalities of different metrics.

Established limit relations between sharp constants for polynomials and entire functions.

Discussed relations in univariate weighted spaces.

Abstract

We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in $\R^m$ and the corresponding constants for entire functions of spherical type on $\R^m$. Certain relations in the univariate weighted spaces are discussed as well.

Equations

$$|{\Delta }^{N}P(0)|\le C_n^{\ast }{\left({\int }_{|x|\le 1}|P(x){|}^{p}dx\right)}^{1/p}$$

$$\Vert {\Delta }^{N}f{\Vert }_{L_{\infty }({\mathbb{R}}^m)}\le C^{\ast }{\left({\int }_{{\mathbb{R}}^m}|f(x){|}^{p}dx\right)}^{1/p}$$

$${\left({\int }_{|x|\le 1}|\tilde{P}(x){|}^{p}dx\right)}^{1/p}\le {\left({\int }_{|x|\le 1}|P(x){|}^{p}dx\right)}^{1/p},p\in [1,\infty ].$$

$${\mathcal{C}}_a={\mathcal{C}}_a\left(\mathcal{B},F({\Omega }_m),D\right):=\underset{f\in \mathcal{B}\setminus \{0\}}{\sup }\frac{\left|D(f)(a)\right|}{\Vert f{\Vert }_{F({\Omega }_m)}}.$$

$${\mathcal{C}}_a\left(B,F({\Omega }_m),D\right)={\mathcal{C}}_a\left(B\cap F({\Omega }_m{)}^{G_m},F({\Omega }_m),D\right).$$

$$\frac{\left|D(f)(a)\right|}{\Vert f{\Vert }_{F({\Omega }_m)}}\le \underset{f^{\ast }\in \left(B\cap F({\Omega }_m{)}^{G_m}\right)\setminus \{0\}}{\sup }\frac{\left|D(f^{\ast })(a)\right|}{\Vert f^{\ast }{\Vert }_{F({\Omega }_m)}}$$

$$f^{\ast }(\cdot ):={\int }_{G_m}f(s\cdot )d\mu (s)$$

$$f^{\ast }(t\cdot )={\int }_{G_m}f(st\cdot )d\mu (s)={\int }_{G_m}f(s\cdot )d\mu (s)=f^{\ast }(\cdot ),$$

$$\Vert f^{\ast }{\Vert }_{F({\Omega }_m)}\le {\int }_{G_m}\Vert f(s\cdot ){\Vert }_{F({\Omega }_m)}d\mu (s)=\Vert f{\Vert }_{F({\Omega }_m)}.$$

$$D(f^{\ast })(x)={\int }_{G_m}D(f_s)(x)d\mu (s)={\int }_{G_m}D(f)(sx)d\mu (s),x\in {\Omega }_m,$$

$$D(f^{\ast })(a)=D(f)(a).$$

$$\frac{\left|D(f)(a)\right|}{\Vert f{\Vert }_{F({\Omega }_m)}}\le \frac{\left|D(f^{\ast })(a)\right|}{\Vert f^{\ast }{\Vert }_{F({\Omega }_m)}}.$$

$$\displaystyle\|f\|_{L_{p}({\Omega_{m}})}:=\left\{\begin{array}[]{ll}\left(\int_{\Omega_{m}}|f(x)|^{p}dx\right)^{1/p},&1\leq p<{\infty},\\ \mbox{ess}\sup_{x\in{\Omega_{m}}}|f(x)|,&p={\infty},\end{array}\right.$$

$$\displaystyle\|f\|_{L_{p}(S^{m-1})}:=\left\{\begin{array}[]{ll}\left(\int_{S^{m-1}}|f(x)|^{p}dS(x)\right)^{1/p},&1\leq p<{\infty},\\ \mbox{ess}\sup_{x\in S^{m-1}}|f(x)|,&p={\infty},\end{array}\right.$$

$$\displaystyle\|f\|_{L_{p,\mu(t)}(\Omega_{1})}:=\left\{\begin{array}[]{ll}\left(\int_{\Omega_{1}}|f(t)|^{p}\mu(t)dt\right)^{1/p},&0<p<{\infty},\\ \mbox{ess}\sup_{t\in\Omega_{1}}|f(t)|,&p={\infty}.\end{array}\right.$$

$${\Vert f+g\Vert }_{L_{p,\mu (t)}({\Omega }_1)}^{\tilde{p}}\le {\Vert f\Vert }_{L_{p,\mu (t)}({\Omega }_1)}^{\tilde{p}}+{\Vert g\Vert }_{L_{p,\mu (t)}({\Omega }_1)}^{\tilde{p}},$$

$$|f(z)|\le C_0(\epsilon ,f)\exp (\sigma (1+\epsilon )|z|),z\in {\mathbb{C}}^m,$$

$$\underset{s\to \infty }{\lim }{f}_{n_s}=f_0$$

$$|f(z)|\le C_0(\epsilon ,f)\exp \left(\sigma (1+\epsilon )\sum _{j=1}^m|z_i|\right),z\in {\mathbb{C}}^m.$$

$$|f_n(x+iy)|\le C\exp (\sigma |y|),n\in \mathbb{N},x\in {\mathbb{R}}^m,y\in {\mathbb{R}}^m,$$

$$|f_0(x+iy)|=\underset{s\to \infty }{\lim }|f_{n_s}(x+iy)|\le C\exp (\sigma |y|)\le C\exp (\sigma |z|).$$

$$A_{\gamma }=\left[\begin{array}{cc}\cos 2\gamma & \sin 2\gamma \\ \sin 2\gamma & -\cos 2\gamma \end{array}\right],$$

$$\Delta ={\Delta }_x:=\sum _{j=1}^m\frac{{\partial }^2}{\partial x_j^2}$$

$${\Delta }_x=\frac{{\partial }^2}{\partial r^2}+\frac{m-1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\mathbf{\delta },$$

$$\mathbf{\delta }:=\sum _{j=1}^{m-1}\frac{1}{q_j{\sin }^{m-j-1}{\theta }_j}\frac{\partial }{\partial {\theta }_j}\left({\sin }^{m-j-1}{\theta }_j\frac{\partial }{\partial {\theta }_j}\right)$$

$${\Delta }_x\left(r^l(x,a{)}^k\right)=\left(\frac{k(k-1)}{(x,a{)}^2}+\frac{l(l+m+2k-2)}{r^2}\right)r^l(x,a{)}^k,r=|x|.$$

$$B{e}_{\nu }(\phi )(r):={\phi }^{\prime \prime }(r)+\frac{2\nu +1}{r}{\phi }^{\prime }(r),\nu \ge -1/2,r\in {\mathbb{R}}^1,$$

$$G{e}_{\lambda }\left(\phi \right)(t):=(1-t^2){\phi }^{\prime \prime }(t)-(2\lambda +1)t{\phi }^{\prime }(t),\lambda \ge -1/2,t\in [-1,1],$$

$${\mathbf{\delta }}_x((x,a{)}^k)={\Delta }_x(r^{-k}(x,a{)}^k)=k(k-1)(x,a{)}^{k-2}-k(k+m-2)(x,a{)}^k=G{e}_{m/2-1}(\phi )(t)|{}_{t=(x,a)}.$$

$$P|{}_{{\Omega }_m}(sx)=P(sx)=P(s\cdot )|{}_{{\Omega }_m}(x).$$

Full text

1. Introduction

We continue the study of the sharp constants in multivariate inequalities of approximation theory that began in [17]. In this paper we prove invariance theorems for multivariate inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials and entire functions of exponential type. In addition, we discuss the asymptotic behavior of certain sharp constants in univariate weighted spaces.

Notation. Let ${\mathbb{R}}^{m},\,m\geq 1,$ be the Euclidean $m$-dimensional space with elements $x=(x_{1},\ldots,x_{m}),\linebreak y=(y_{1},\ldots,y_{m})$, the inner product $(x,y):=\sum_{j=1}^{m}x_{j}y_{j}$, and the norm $|x|:=\sqrt{(x,x)}$. Next, ${\mathbb{C}}^{m}:={\mathbb{R}}^{m}+i{\mathbb{R}}^{m}$ is the $m$-dimensional complex space with elements $z=x+iy=(z_{1},\ldots,z_{m})$ and the norm $|z|:=\sqrt{\sum_{j=1}^{m}|z_{j}|^{2}}=\sqrt{|x|^{2}+|y|^{2}}$; ${\mathbb{Z}}^{m}_{+}$ denotes the set of all integral lattice points in ${\mathbb{R}}^{m}$ with nonnegative coordinates. In addition, we use a multi-index ${\alpha}=({\alpha}_{1},\ldots,{\alpha}_{m})\in{\mathbb{Z}}^{m}_{+}$ with $|{\alpha}|:=\sum_{j=1}^{m}{\alpha}_{j}$ and $x^{\alpha}:=x_{1}^{{\alpha}_{1}}\cdot\cdot\cdot x_{m}^{{\alpha}_{m}}$. We also use the Frobenius norm $\|A\|_{F}:=\left(\sum_{k,j=1}^{m}a_{k,j}^{2}\right)^{1/2}$ of an $m\times m$ matrix $A=\left[a_{k,j}\right]_{k,j=1}^{m}$ with real elements. Given $M>0$, let ${\mathfrak{B}}^{m}(M):=\{x\in{\mathbb{R}}^{m}:|x|\leq M\},\,{\mathfrak{B}}^{m}:={\mathfrak{B}}^{m}(1)$, and $S^{m-1}:=\{x\in{\mathbb{R}}^{m}:|x|=1\}$ be the $m$-dimensional ball of radius $M$, the unit $m$-dimensional ball, and the unit $(m-1)$-dimensional sphere in ${\mathbb{R}}^{m}$, respectively. Next, let $|E|_{k}$ denote the $k$-dimensional Lebesgue measure of a $k$-dimensional measurable set $E\subseteq{\mathbb{R}}^{m}$. In particular, $|S^{m-1}|_{m-1}=2\pi^{m/2}/\Gamma(m/2),\,m\geq 2$, and $|S^{0}|_{0}:=2$. In addition, we use generic notation $\lfloor a\rfloor,\,\Gamma(z)$ and $\mbox{B}({\alpha},{\beta})$ for the floor function, the gamma function and the beta function, respectively.

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.

** Markov-Bernstein-Nikolskii Type Inequalities.** Limit relations between sharp constants in the univariate Markov-Bernstein-Nikolskii type inequalities for trigonometric and algebraic polynomials and entire functions of exponential type were studied by Taikov [34, 35], Gorbachev [21], Levin and Lubinsky [25, 26], the author and Tikhonov [20], and the author [16]. Detailed surveys of the univariate Markov-Bernstein-Nikolskii type inequalities for trigonometric and algebraic polynomials and entire functions of exponential type were presented in [20, 16]. The corresponding multivariate problems were recently studied by Dai, Gorbachev, and Tikhonov [9] and the author [17]. The following publications [24, 3, 22] are also closely related to the subject of the paper.

The purpose of this paper is threefold. First, we extend invariance theorems of approximation theory, proved by the author and Pichugov [19] and by the author [15], to the generalized Markov-Bernstein-Nikolskii type inequalities. These results are presented and proved in Section 2 (see Theorems 2.1 and 2.2). In particular, invariance theorems reduce certain multivariate inequalities to univariate ones in weighted metrics. Certain special cases are discussed in Section 3 (see Examples 3.8, 3.10, 3.12 and Corollaries 3.9, 3.11, 3.13).

Second, in Section 4 we obtain limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for polynomials on the unit sphere and the unit ball in ${\mathbb{R}}^{m}$ and the corresponding constants for entire functions of spherical type on ${\mathbb{R}}^{m}$ (see Corollaries 4.4 and 4.5). In particular, the major part of Corollary 4.4 is the following statement: if $\Delta$ is the Laplace operator, then the minimal (sharp) constant $C_{n}^{*}$ in the inequality

[TABLE]

over all polynomials $P$ of degree at most $n$ in $m$ variables and the minimal constant $C^{*}$ in the inequality

[TABLE]

($p\in[1,{\infty}],\,N=0,\,1\,\ldots$) over all entire functions $f$ of spherical type $1$ are connected by the limit relation $\lim_{n\to{\infty}}n^{-2N-m/p}C_{n}^{*}=C^{*}.$

A similar result on the unit sphere is proved in Corollary 4.5. A special case of Corollary 4.5 for $N=0$ was established in [9].

The proofs of multivariate results are based on the invariance theorems and certain univariate results. By using the invariance theorems, the limit multivariate relations can be reduced to the relations between sharp constants in the univariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials with the Bessel and Gegenbauer differential operators in weighted $L_{p}$-spaces on $[-1,1],\,1\leq p\leq{\infty}$, and the corresponding constants for univariate entire functions of exponential type.

Third, limit relations between the univariate sharp constants in more general weighted spaces are presented in Theorems 4.1 and 4.3. Their proofs are given in Section 6. Note that the proofs are based on an approach to limit relations between sharp constants developed in [20, 16, 17].

Surprisingly, a special case of limit relations between the univariate sharp constants in weighted spaces is the asymptotic relation for the sharp constant in the classical inequality for univariate polynomials of different metrics (see Corollary 4.6; cf. [16, Theorem 1.4 and p. 94]).

Special cases of the results in Section 4 are obtained earlier in [25, 16, 9]. Section 5 contains certain properties of entire functions of exponential type and polynomials that are needed for the proofs.

2. General Invariance Theorems

The averaging (or symmetrization) principle is well known in approximation theory (see, e.g., [27, 8, 2, 19, 15] and references therein). In particular, general and special invariance theorems for the error of best approximation were proved in [19, 15]. In this section we discuss general invariance theorems for the sharp constants in the Markov-Bernstein-Nikolskii type inequalities.

It is easy to see that the sharp constant $C_{n}^{*}$ in inequality (1.1) for $p\in[1,{\infty}]$ does not change if one restricts (1.1) to even polynomials $P$ (i.e., $P(x)=P(-x),\,|x|\leq 1).$ The proof of this fact involves first, the symmetrization operator $\tilde{P}(x):=(P(x)+P(-x))/2$ of $P$ and second, the invariance of the operator $\Delta^{N}$ and the class ${\mathcal{P}}_{n,m}$ under the group $G_{m}=\{-e^{*},e^{*}\}$, where $e^{*}$ is the identity transformation on the unit ball ${\Omega_{m}}$. Indeed, $|\Delta^{N}\tilde{P}(0)|=|\Delta^{N}P(0)|$ and

[TABLE]

Below we extend this example to compact topological groups $G_{m}$ of continuous transformations on more general sets ${\Omega_{m}}$ and to more general classes of elements. The symmetrization will be provided by the Haar integral. More examples are discussed in Section 3.

Let ${\Omega_{m}}\subseteq{\mathbb{R}}^{m}$ and let $F({\Omega_{m}})$ be a Banach space of functions $f:{\Omega_{m}}\to{\mathbb{C}}^{1}$ with the norm $\|\cdot\|_{F({\Omega_{m}})}$. Next, let ${\mathcal{B}}\neq\{0\}$ be a closed subspace of $F({\Omega_{m}})$ and let $D:{\mathcal{B}}\to F({\Omega_{m}})$ be a bounded linear operator. Given $a\in{\Omega_{m}}$, we define the sharp constant in the generalized Markov-Bernstein-Nikolskii type inequality by

[TABLE]

If $D$ is the imbedding operator $I:{\mathcal{B}}\to F({\Omega_{m}})$, then ${\mathcal{C}}_{a}$ is the sharp constant in the generalized Nikolskii-type inequality.

Further, let $G_{m}=G_{m}(a)$ be a compact topological group of continuous transformations $s:{\Omega_{m}}\to{\Omega_{m}}$ with a fixed point $a\in{\Omega_{m}}$ (i. e., $sa=a,\,s\in G_{m}$) and let $F({\Omega_{m}})^{G_{m}}$ denote a subspace of $F({\Omega_{m}})$ of all functions $f$ which are invariant under the group $G_{m}$, i.e., $f(s\cdot)=f(\cdot),\,s\in G_{m}$.

Let $B$ be a closed subspace of $F({\Omega_{m}})$. In this section we discuss sufficient conditions for the sharp constant to be invariant under $G_{m}$, i.e.,

[TABLE]

We assume that $B,\,G_{m},\,D,$ and $F({\Omega_{m}})$ satisfy the following conditions.

• (C1)

The norm $\|\cdot\|_{F({\Omega_{m}})}$ is invariant under $G_{m}$, i.e., for every $f\in F({\Omega_{m}})$ and each $s\in G_{m},\,\|f(s\cdot)\|_{F({\Omega_{m}})}=\|f\|_{F({\Omega_{m}})}$.

• (C2)

The operator $D$ is invariant under $G_{m}$, i.e., for every $f\in B$ and each $s\in G_{m},\,D(f_{s})(\cdot)=D(f)(s\cdot)$, where $f_{s}(\cdot):=f(s\cdot)$.

• (C3)

The subspace $B$ is invariant under $G_{m}$, i.e., for every $f\in B$ and each $s\in G_{m},\,f(s\cdot)\in B$.

• (C4)

For every $f\in B$ the mapping $f(s\cdot):G_{m}\to B$ is a continuous function in $s\in G_{m}$, i.e., for any $s\in G_{m},\,\lim_{\tau\to s}\|f(s\cdot)-f(\tau\cdot)\|_{F({\Omega_{m}})}=0$.

• (C5)

$B\cap F({\Omega_{m}})^{G_{m}}\neq\{0\}$.

The following general invariance theorem holds true.

Theorem 2.1**.**

If conditions (C1) through (C5) are satisfied, then (2.2) is valid.

Proof.

It suffices to prove the inequality

[TABLE]

for every $f\in B\setminus\{0\}$. Note that due to condition (C5) the right-hand sides of (2.2) and (2.3) are well-defined.

Since $G_{m}$ is a compact topological group, there exists the Haar measure $\mu(s)$ on $G_{m}$ with $\mu(G_{m})=1$ (see, e.g., [31, Theorem 5.14]). Next, for every $f\in B$ the function $f(s\cdot):G_{m}\to B$ is continuous on $G_{m}$ by conditions (C3) and (C4); therefore, its image $H:=\{f(s\cdot):s\in G_{m}\}$ is compact in $F({\Omega_{m}})$. Since $F({\Omega_{m}})$ is a Banach space, the closure of the convex hull of $H$ (denote it by $\bar{H}_{co}$) is compact as well (see, e.g., [31, Theorem 3.25 (a)]). Then the Haar integral

[TABLE]

exists and $f^{*}\in\bar{H}_{co}$ (see, e.g. [31, Theorem 3.27]). Note that examples of the Haar measures and the Haar integral can be found in [23, Sect. 15.17] and [32, Sect. 12.1]. Moreover, since $B$ is a closed subset of $F({\Omega_{m}})$, we conclude that $\bar{H}_{co}\subseteq B$, so $f^{*}\in B$. Next, for every $t\in G_{m}$,

[TABLE]

where the second equality in (2.5) follows from the invariance of the Haar measure. Therefore, $f^{*}\in B\cap F({\Omega_{m}})^{G_{m}}$. Using now the generalized Minkowski inequality (see, e.g., [10, Lemma 3.2.15]) and condition (C1), we obtain

[TABLE]

Further, by [10, Theorem 3.2.19(c)] and conditions (C3) and (C2), we have

[TABLE]

and since $sa=a,\,s\in G_{m}$, we obtain from (2.7)

[TABLE]

If $f\neq 0$ and $D(f)(a)=0$, then (2.3) holds trivially true by (C5). If $D(f)(a)=D(f^{*})(a)\neq 0$, then $f\neq 0,\,f^{*}\neq 0$, and it follows from (2.6) and (2.8) that

[TABLE]

Hence (2.3) holds true in this case as well. Thus (2.2) is established. $\Box$

The proof of Theorem 2.1 is based on the existence of the Haar integral $f^{*}$ which belongs to $B$. These both facts follow from strong condition (C4). However, the existence of $f^{*}(x)$ for each $x\in{\Omega_{m}}$ follows from the following weaker condition.

• (C4′)

For every $f\in B$ and each fixed $x\in{\Omega_{m}}$, the linear functional $f(sx):G_{m}\to{\mathbb{C}}^{1}$ is continuous in $s\in G_{m}$.

It is obvious that (C4) implies (C4′), but the converse statement is not valid in general (see [15, Example 2.1]). If we introduce a new condition

• (C6)

For every $f\in B$, the Haar integral $f^{*}$ defined in (2.4) belongs to $B$,

then we arrive at the following version of Theorem 2.1.

Theorem 2.2**.**

If conditions (C1), (C2), (C3), (C4′), (C5), and (C6) are satisfied, then (2.2) is valid.

Proof.

The existence of $f^{*}$ follows from (C4′) (see, e.g., [31, Theorem 5.14]) and, in addition, $f^{*}\in B$ by (C6). The rest of the proof of (2.2) is similar to that of Theorem 2.1. $\Box$

Note that invariant means like the Haar integral exist in more general situation, more precisely, on amenable semigroups (see [2, Sect. 6] for details).

3. Special Invariance Theorems

Here, we discuss special cases of invariance theorems presented in Section 2.

Special Cases and Preliminaries. In all our examples of applications of Theorems 2.1 and 2.2 we use special sets ${\Omega_{m}}$, spaces $F({\Omega_{m}})$, subspaces $B$, groups $G_{m}$, and linear operators $D$ described below. In addition, we discuss here their certain properties.

Let ${\Omega_{m}}$ be one of the following sets: ${\mathfrak{B}}^{m},\,m\geq 1;\,S^{m-1},\,m\geq 2;$ and ${\mathbb{R}}^{m},\,m\geq 1$; and let $F({\Omega_{m}})=L_{p}({\Omega_{m}}),\,1\leq p\leq{\infty}$, be the space of all measurable functions $f:{\Omega_{m}}\to{\mathbb{C}}^{1}$ with the finite norm

[TABLE]

if ${\Omega_{m}}={\mathfrak{B}}^{m}$ or ${\Omega_{m}}={\mathbb{R}}^{m}$ and

[TABLE]

where $S(\cdot)$ is the spherical surface Lebesgue measure on $S^{m-1}$. In addition, we also need the weighted space $L_{p,\mu(t)}(\Omega_{1}),\,0<p\leq{\infty},$ of all univariate measurable functions $f:\Omega_{1}\to{\mathbb{C}}^{1}$ with the finite quasinorm

[TABLE]

Here, $\Omega_{1}$ is a measurable subset of ${\mathbb{R}}^{1}$ and $\mu:\Omega_{1}\to[0,{\infty})$ is a locally integrable weight. This quasinorm allows the following ”triangle” inequality

[TABLE]

where $\tilde{p}:=\min\{1,p\}$ for $p\in(0,{\infty}]$. In this section, $\mu(t)$ is either $|t|^{m-1},\,m\geq 1$, or $(1-t^{2})^{(m-3)/2},\,m\geq 2$. In Sections 4, 5, and 6, we use more general weights.

In the capacity of $B$ we discuss either the set ${\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.$ of the restrictions $P\left|{}_{{\Omega_{m}}}\right.$ to ${\Omega_{m}}$ of polynomials $P(x)=\sum_{|{\alpha}|\leq n}c_{\alpha}x^{\alpha}\,:{\mathbb{R}}^{m}\to{\mathbb{C}}^{1}$ in $m$ variables of degree at most $n$ (if ${\Omega_{m}}$ is identified, we often write ${\mathcal{P}}_{n,m}$ instead of ${\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.$) or the set $E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})$ of the restrictions to ${\mathbb{R}}^{m}$ of entire functions in $m$ variables of spherical type ${\sigma}>0$ that belong to $L_{p}({\mathbb{R}}^{m})$.

We recall that an entire function $f:{\mathbb{C}}^{m}\to{\mathbb{C}}^{1}$ has spherical type ${\sigma}>0$ if for any ${\varepsilon}>0$ there exists a constant $C_{0}({\varepsilon},f)$ such that

[TABLE]

(see [29, Sect. 3.2.6] and [11, Definition 5.1]). We often identify $f$ with its restriction to ${\mathbb{R}}^{m}$. Note that $E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})$ is a closed subspace of $L_{p}({\mathbb{R}}^{m}),\,p\in(0,{\infty}]$ (cf. [29, Theorem 3.5]).

We need the following compactness theorem for functions from $E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})$.

Proposition 3.1**.**

For any sequence $\{f_{n}\}_{n=1}^{\infty},\,f_{n}\in E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m}),\,n\in{\mathbb{N}},\,p\in[1,{\infty}]$, with $\sup_{n\in{\mathbb{N}}}\|f_{n}\|_{L_{\infty}({\mathbb{R}}^{m})}=C$, there exist a subsequence $\{f_{n_{s}}\}_{s=1}^{\infty}$ and a function $f_{0}\in E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})$ such that

[TABLE]

uniformly on any compact set in ${\mathbb{C}}^{m}$.

Proof.

It follows from (3.5) that if $f\in E_{{\sigma},m}$, then

[TABLE]

Therefore, $f_{n},\,n\in{\mathbb{N}},$ has exponential type ${\sigma}$ by the definition in [29, Sect. 3.1]. Since $f_{n}\in E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})$, by Nikolskii’s compactness theorem [29, Theorem 3.3.6], there exists a subsequence $\{f_{n_{s}}\}_{s=1}^{\infty}$ and an entire function $f_{0}\in L_{\infty}({\mathbb{R}}^{m})$ such that (3.6) holds true uniformly on any compact set in ${\mathbb{C}}^{m}$. In addition, $f_{0}\in E_{{\sigma},m}.$ Indeed, since

[TABLE]

(see [28, Eq. (4.13)]), we obtain by (3.8)

[TABLE]

Thus $f_{0}\in E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})$. $\Box$

In addition to ${\mathcal{P}}_{n,m}={\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.$ and $E_{{\sigma},m}$, we also need univariate sets ${\mathcal{P}}_{n,1,e}={\mathcal{P}}_{n,1,e}\left|{}_{[-1,1]}\right.$ and $E_{{\sigma},1,e}$ of all even polynomials and even entire functions from ${\mathcal{P}}_{n,1}={\mathcal{P}}_{n,1}\left|{}_{[-1,1]}\right.$ and $E_{{\sigma},1}$, respectively.

Throughout the paper we use the following groups $G_{m}=G_{m}(a)$. Let $G_{m}(0)=O(m)$ be the group of all proper and improper rotations (about the origin) of ${\mathbb{R}}^{m}$. We identify $O(m)$ with the group of all $m\times m$ orthogonal matrices which is isomorphic to $O(m)$ since $s\in O(m)$ if and only if $sx=A(s)\,x^{T}$, where $A(s)$ is an $m\times m$ orthogonal matrix with $|\det A(s)|=1$ and $x^{T}$ is a column vector. Let $G_{m}(a)=O(m,a)$ be a subgroup of $O(m)$ of all proper and improper rotations (or $m\times m$ orthogonal matrices) $s$, satisfying the condition $sa=a$, where $a\neq 0$ is a fixed vector from ${\mathbb{R}}^{m}$. For example, if $a=(\cos{\gamma},\sin{\gamma})$, then $O(2,a)=\{I,A_{\gamma}\}$, where $I$ is the $2\times 2$ identity matrix and

[TABLE]

which is the product of two transformations: reflection about the $x$-axis and rotation by the angle $2{\gamma}$. Note that $O(m)$ will be used in Examples 3.8 and 3.10, while $O(m,a)$ will be used in Example 3.12.

Throughout the paper we use the polyharmonic operator $D=\Delta^{N}$, where $N\in{\mathbb{Z}}^{1}_{+}$ and

[TABLE]

is the Laplace operator on ${\Omega_{m}}$. Note that the restriction $\boldsymbol{{\delta}}_{x}$ of $\Delta_{x}$ on $S^{m-1}$ is defined by $\boldsymbol{{\delta}}_{x}(f)(x)=\Delta_{x}(f)\left|{}_{S^{m-1}}\right.(x):=\Delta_{x}(f)(x/|x|)$. In case of $N=0,\,\Delta^{N}$ is the imbedding operator $I:{\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.\to L_{p}({\Omega_{m}})$ or $I:E_{{\sigma},m}\cap L_{p}({\mathbb{R}}^{m})\to L_{p}({\mathbb{R}}^{m})$. We need certain properties of $\Delta$.

Proposition 3.2**.**

*Let ${\mathbb{R}}^{m},\,m\geq 2$, be equipped with the spherical coordinates $x=(r,\,\theta_{1},\ldots,\theta_{m-1}),\,r\in[0,{\infty}),\,\theta_{j}\in[0,\pi],\,1\leq j\leq m-2,\,\theta_{m-1}\in[0,2\pi).$ Then the following properties of $\Delta$ hold true.

(a) In spherical coordinates,*

[TABLE]

where $\boldsymbol{{\delta}}$ is the spherical Laplacian given by

[TABLE]

*and $q_{1}:=1;q_{j}:=\prod_{s=1}^{j-1}\sin^{2}\theta_{s},\,2\leq j\leq m-1$. In particular, $\boldsymbol{{\delta}}$ is the representation of $\boldsymbol{{\delta}}_{x}$ in spherical coordinates.

(b) For a fixed $a\in S^{m-1},\,l\in{\mathbb{R}}^{1},$ and $k\in{\mathbb{R}}^{1},$*

[TABLE]

(c) If $f(x)$ is a radial function ${\varphi}(|x|)$, where ${\varphi}:{\mathbb{R}}^{1}\to{\mathbb{C}}^{1}$ is an even twice continuously differentiable function on ${\mathbb{R}}^{1}$, then $\Delta_{x}(f)(x)=Be_{m/2-1}\left({\varphi}\right)(|x|),\,m\geq 1$. Here,

[TABLE]

*is the Bessel operator and $Be_{\nu}({\varphi})(0):=\lim_{r\to 0}Be_{\nu}({\varphi})(r)$.

(d) If $f(x)={\varphi}((x,a))$, where ${\varphi}\in{\mathcal{P}}_{n,1},\,x\in S^{m-1}$, and $a\in S^{m-1}$ is a fixed point, then $\boldsymbol{{\delta}}_{x}(f)(x)=Ge_{m/2-1}\left({\varphi}\right)((x,a)),\,m\geq 2$. Here,*

[TABLE]

*is the Gegenbauer operator.

(e) The operators $\Delta_{x}$ and $\boldsymbol{{\delta}}_{x}$ are invariant under orthogonal transformations, i.e., for $s\in O(m)$ and $y=sx,\,\Delta_{x}=\Delta_{y}$ and $\boldsymbol{{\delta}}_{x}=\boldsymbol{{\delta}}_{y}$.*

Proof.

Statements (a), (b), and (e) are well-known and can be found in [12, Sect. 11.1.1], while (c) follows immediately from (a). It suffices to prove (d) for ${\varphi}(t)=t^{k},\,k=0,\,1,\,\ldots,n.$ Using statement (b) for $l=-k$ and $r=1$, we obtain

[TABLE]

This completes the proof of the proposition. $\Box$

Remark 3.3*.*

We call (3.9) the Bessel operator because the functions $t^{-\nu}J_{\nu}(\sqrt{c}t)$ are eigenfunctions of $Be_{\nu}$ for $c\geq 0$, see [37, Sect. 4.31]. We call (3.10) the Gegenbauer operator because the Gegenbauer polynomials $C_{k}^{{\lambda}}$ are eigenfunctions of $Ge_{{\lambda}}$, see [12, Sect. 10.9].

Conditions in Special Cases. We first discuss conditions (C1), (C2), and (C3) in special cases.

Proposition 3.4**.**

*Let $G_{m}=G_{m}(a)$ be a subgroup of $O(m,a)$. Then the following statements hold true.

(a) If ${\Omega_{m}}$ is one of the sets ${\mathfrak{B}}^{m},\,m\geq 1;\,S^{m-1},m\geq 2;\,{\mathbb{R}}^{m},\,m\geq 1$, and $F({\Omega_{m}})=L_{p}({\Omega_{m}}),\,1\leq p\leq{\infty}$, then (C1) is satisfied.

(b) If $D$ is the polyharmonic operator $\Delta^{N},\,N=0,\,1,\dots,$ then (C2) is satisfied.

(c) If $B={\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.$, where either ${\Omega_{m}}={\mathfrak{B}}^{m},\,m\geq 1$, or ${\Omega_{m}}=S^{m-1},\,m\geq 2$, then (C3) is satisfied.

(d) If $B=E_{1,m}\cap L_{p}({\mathbb{R}}^{m})$, then (C3) is satisfied.*

Proof.

Statement (a) is obviously satisfied, while (b) follows from Proposition 3.2 (e). Let $P\in{\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.$ and $x\in{\Omega_{m}}$. Since $sx\in{\Omega_{m}}$, we have

[TABLE]

Next, $s$ is a linear transformation, so $P(s\cdot)\in{\mathcal{P}}_{n,m}\left|{}_{{\mathbb{R}}^{m}}\right.$ and $P\left|{}_{{\Omega_{m}}}\right.(s\cdot)\in{\mathcal{P}}_{n,m}\left|{}_{{\Omega_{m}}}\right.$ by (3.11).

Further, let $f\in E_{1,m}\cap L_{p}({\mathbb{R}}^{m})$. Extending $f$ and $f(s\cdot)$ to ${\mathbb{C}}^{m}$, we see that $f(s\cdot)$ is an entire function since $s\in O(m)$ is a linear transformation. Moreover, for $z\in{\mathbb{C}}^{m},\,|sz|=|z|$. Therefore, $f(s\cdot)\in E_{1,m}$ by (3.5). In addition, $f(s\cdot)\in L_{p}({\mathbb{R}}^{m})$ by statement (a). Thus statements (c) and (d) are established. $\Box$

In the following two propositions we discuss the validity of conditions (C4) and (C4′) in special cases.

Proposition 3.5**.**

*Let ${\Omega_{m}}$ be a closed subset of ${\mathbb{R}}^{m}$ and $F({\Omega_{m}})=L_{p}({\Omega_{m}}),\,1\leq p\leq{\infty}$. Next, let $B$ be a subspace of $L_{p}({\Omega_{m}})$ of continuous functions on ${\Omega_{m}}$, and let $G_{m}^{*}$ be a compact group of linear transformations of the form $sx=A(s)\,x^{T}:{\Omega_{m}}\to{\Omega_{m}}$, where $A(s)$ is an $m\times m$ matrix. Then the following two statements hold true.

(a) Condition (C4′) is satisfied for $1\leq p\leq{\infty}$.

(b) Condition (C4) is satisfied for $1\leq p\leq{\infty}$ and a compact set ${\Omega_{m}}$.*

Proof.

Statement (a) follows from the uniform continuity of $f(sx)$ in $s\in G_{m}^{*}$ for each $x\in{\Omega_{m}}$ if we take into account the elementary inequality

[TABLE]

Similarly, statement (b) follows from the uniform continuity of $f(s\cdot)$ in $s\in G_{m}^{*}$, the estimate

[TABLE]

and the continuous imbedding of $L_{\infty}({\Omega_{m}})$ into $L_{p}({\Omega_{m}})$ for a compact set ${\Omega_{m}}$. $\Box$

Proposition 3.6**.**

Let $B$ be a subspace of $L_{p}({\mathbb{R}}^{m}),\,1\leq p<{\infty},$ of continuous functions on ${\mathbb{R}}^{m}$, and let $G_{m}^{*}$ be a subgroup of $O(m)$. Then condition (C4) is satisfied.

Proof.

Let $f\in B$ and let $s$ and $s_{1}$ be two proper or improper rotations and let $A(s)$ and $A(s_{1})$ be the corresponding orthogonal matrices. Given ${\varepsilon}>0$ there exists $M=M({\varepsilon},f)>0$ such that

[TABLE]

Next, by statement (b) of Proposition 3.5, there exists ${\delta}({\varepsilon})>0$ such that for $\|s-s_{1}\|:=\|A(s)-A(s_{1})\|_{F}<{\delta}({\varepsilon})$,

[TABLE]

Combining (3.12) with (3.13), we obtain

[TABLE]

Then condition (C4) is satisfied. $\Box$

Remark 3.7*.*

Note that condition (C4) is not always satisfied if $B$ is a subspace of $L_{\infty}({\mathbb{R}}^{m})$ of continuous functions on ${\mathbb{R}}^{m}$, see [15, Example 2.1].

Examples. Here, we discuss typical examples of applications of Theorems 2.1 and 2.2. In addition to $D=\Delta^{N}$, we also use Bessel and Gegenbauer operators $D=(Be_{\nu})^{N}$ and $D=(Ge_{\lambda})^{N}$ defined by (3.9) and (3.10). In case of $N=0,\,(Be_{\nu})^{N}$ and $(Ge_{\lambda})^{N}$ are the corresponding imbedding operators.

Example 3.8**.**

${\Omega_{m}}={\mathfrak{B}}^{m},\,F({\Omega_{m}})=L_{p}({\mathfrak{B}}^{m}),\,m\geq 1,\,1\leq p\leq{\infty};\,G_{m}=G_{m}(0)=O(m);\linebreak L_{p}({\mathfrak{B}}^{m})^{O(m)}$ is the set of all radial functions from $L_{p}({\mathfrak{B}}^{m});\,B={\mathcal{P}}_{n,m};\,D=\Delta^{N}=\Delta^{N}\left|{}_{{\mathfrak{B}}^{m}}\right.,\linebreak N\in{\mathbb{Z}}^{1}_{+}$.

Conditions (C1), (C2), and (C3) are satisfied by Proposition 3.4 and (C4) is satisfied by Proposition 3.5 (b). In addition, for $1\leq p\leq{\infty}$,

[TABLE]

(see [33, Lemma 4.2.11] and [15, Proposition 4.1]). In particular, condition (C5) is satisfied by (3.14).

Using (3.14) and Proposition 3.2 (c), we obtain from Theorem 2.1

[TABLE]

Thus we arrive at the following corollary.

Corollary 3.9**.**

For $m\in{\mathbb{N}},\,n\in{\mathbb{N}},\,N\in{\mathbb{Z}}^{1}_{+}$, and $p\in[1,{\infty}]$,

[TABLE]

Example 3.10**.**

${\Omega_{m}}={\mathbb{R}}^{m},\,F({\Omega_{m}})=L_{p}({\mathbb{R}}^{m}),\,m\geq 1,\,1\leq p\leq{\infty};\,G_{m}=G_{m}(0)=O(m);\linebreak L_{p}({\mathbb{R}}^{m})^{O(m)}$ is the set of all radial functions from $L_{p}({\mathbb{R}}^{m});\,B=E_{1,m}\cap L_{p}({\mathbb{R}}^{m});\,D=\Delta^{N},\,N\in{\mathbb{Z}}^{1}_{+}$.

Conditions (C1), (C2), and (C3) are satisfied by Proposition 3.4 and condition (C4) for $p\in[1,{\infty})$ is satisfied by Proposition 3.6.

In case of $p={\infty}$, condition (C4′) is satisfied by Proposition 3.5 (a). Next, we show that condition (C6) is satisfied for $p={\infty}$ and condition (C5) is satisfied for $1\leq p\leq{\infty}$.

We first state the following simple fact (cf. [29, Lemma 3.6.1]) that is used in this example: if $h(t)\in E_{{\sigma},1}$ is an even function, then ${\varphi}(x):=h(|x|)\in E_{{\sigma},m}$. Indeed, it is an entire function with the extension $h(|z|)$ to ${\mathbb{C}}^{m}$. Moreover, since $|h(|z|)|\leq C_{0}({\varepsilon},h)\exp({\sigma}(1+{\varepsilon})|z|)$ by(3.5), we conclude that ${\varphi}\in E_{{\sigma},m}$.

In particular, the function

[TABLE]

belongs to $E_{1/n,m},\,n\in{\mathbb{N}}.$

Now we show that condition (C6) is satisfied, i.e., for $f\in E_{1,m}\cap L_{\infty}({\mathbb{R}}^{m})$ the Haar integral

[TABLE]

belongs to $E_{1,m}\cap L_{\infty}({\mathbb{R}}^{m})$. Let us set $f_{n}(x):=f(x)h_{n}(x),\,n\in{\mathbb{N}}$. Then $f_{n}\in E_{1+1/n,m}\cap L_{1}({\mathbb{R}}^{m}),\,n\in{\mathbb{N}}$, and by the elementary inequality $1-\sin t/t\leq t^{2}/6,\,t\in R^{1}$, we obtain

[TABLE]

Next, for the corresponding Haar integral we have

[TABLE]

We prove (3.17) similarly to the fact that $f^{*}\in B$ in the proof of Theorem 2.1. The mapping $f_{n}(s\cdot):O(m)\to E_{1+1/n,m}\cap L_{1}({\mathbb{R}}^{m})$ is continuous on $O(m)$ by condition (C3) (i.e., $f_{n}\in E_{1+1/n,m}\cap L_{1}({\mathbb{R}}^{m})$; see Proposition 3.4 (d)) and by Condition (C4) (see Proposition 3.5 (b)). Then its image $H:=\{f_{n}(s\cdot):s\in O(m)\}$ is compact in $L_{1}({\mathbb{R}}^{m})$ and the closure of the convex hull $\bar{H}_{co}$ of $H$ is compact as well (see, e.g., [31, Theorem 3.25 (a)]). Then $f_{n}^{*}$ exists and $f_{n}^{*}\in\bar{H}_{co}$; since $\bar{H}_{co}\subseteq E_{1+1/n,m}\cap L_{1}({\mathbb{R}}^{m})$, we conclude that $f_{n}^{*}\in E_{1+1/n,m}\cap L_{1}({\mathbb{R}}^{m})$.

In addition, it follows from (3.16), (3.17), and (3.18)

[TABLE]

Next, (3.19) shows that

[TABLE]

uniformly on any compact set of ${\mathbb{R}}^{m}$. Further,

[TABLE]

Then by Proposition 3.1 and (3.18), there exist a subsequence $\{f^{*}_{n_{s}}\}_{s=1}^{\infty}$ and a function $f_{0}^{*}\in E_{1,m}\cap L_{\infty}({\mathbb{R}}^{m})$ such that

[TABLE]

uniformly on any compact set of ${\mathbb{R}}^{m}$. Comparing (3.20) and (3.21), we conclude that $f^{*}=f_{0}^{*}$, so $f^{*}\in E_{1,m}\cap L_{\infty}({\mathbb{R}}^{m})$. Thus condition (C6) is satisfied in the case $p={\infty}$.

In addition, for $1\leq p\leq{\infty}$,

[TABLE]

(see [15, Proposition 6.1]). For example, the function

[TABLE]

belongs to $E_{1,m}\cap L_{p}({\mathbb{R}}^{m})$. Then condition (C5) is satisfied for $1\leq p\leq{\infty}$.

Thus we can use Theorem 2.1 for $1\leq p<{\infty}$ and use Theorem 2.2 for $p={\infty}$. Finally, taking account of Proposition 3.2 (c) and (3.22), we obtain for $1\leq p\leq{\infty}$

[TABLE]

Thus we arrive at the following corollary.

Corollary 3.11**.**

For $m\in{\mathbb{N}},\,N\in{\mathbb{Z}}^{1}_{+}$, and $p\in[1,{\infty}]$,

[TABLE]

Example 3.12**.**

${\Omega_{m}}=S^{m-1},\,F({\Omega_{m}})=L_{p}(S^{m-1}),\,m\geq 2,\,1\leq p\leq{\infty};\,G_{m}=G_{m}(a)=O(m,a)$, where $a\in S^{m-1}$ is a fixed point; $B={\mathcal{P}}_{n,m}\left|{}_{S^{m-1}}\right.;\,D=\Delta^{N}\left|{}_{S^{m-1}}\right.=\boldsymbol{{\delta}}_{x}^{N}$ (see Proposition 3.2 (a)), $N\in{\mathbb{Z}}^{1}_{+}$.

Conditions (C1), (C2), and (C3) are satisfied by Proposition 3.4 and (C4) is satisfied by Proposition 3.5 (b).

In addition, for $1\leq p\leq{\infty}$,

[TABLE]

(see [15, Proposition 5.2]). In particular, condition (C5) is satisfied by (3.24).

Finally, taking account of the formula (${\varphi}\in L_{p}(S^{m-1})$)

[TABLE]

(see, e.g., [12, Sect. 11.4]) and using (3.24) and Proposition 3.2 (d), we obtain from Theorem 2.1

[TABLE]

Thus we arrive at the following corollary.

Corollary 3.13**.**

For $a\in S^{m-1},\,m\in\{2,\,3,\,\ldots\},\,N\in{\mathbb{Z}}^{1}_{+}$, and $p\in[1,{\infty}]$,

[TABLE]

Note that a similar result for $m\geq 3$ and $N=0$ was proved by Arestov and Deikalova [4, Theorem 2].

4. Univariate and Multivariate Constants

Here, we discuss main results on limit relations between sharp constants in the univariate and multivariate Markov-Bernstein-Nikolskii type inequalities. The necessary notation is introduced in Sections 1, 2, and 3. In particular, the sharp constant ${\mathcal{C}}_{a}$ was defined by (2.1) and $Be_{\nu}$ and $Ge_{\lambda}$ are Bessel and Gegenbauer operators defined by (3.9) and (3.10), respectively.

We first discuss sharp constants in the univariate inequalities of different weighted metrics.

Theorem 4.1**.**

If $\nu\geq-1/2,\,N\in{\mathbb{Z}}^{1}_{+},$ and $p\in(0,{\infty}]$, then the limit relation

[TABLE]

is valid. In addition, there exists a function $f_{0}\in\left(E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})\right)\setminus\{0\}$ such that

[TABLE]

Note that for $\nu=-1/2$ Theorem 4.1 in more general settings was proved in [16, Theorem 1.1].

Remark 4.2*.*

The domain $D_{\nu}(N)$ of the operator $\left(Be_{\nu}\right)^{N}$ is a subset of the set $C^{N}({\mathbb{R}}^{1})$ of all $N$ times differentiable functions on ${\mathbb{R}}^{1}$ that consists of all $h\in C^{N}({\mathbb{R}}^{1})$, satisfying the relations $h^{(l)}(0)\prod_{j=1}^{l}(2j-2\nu+1)=0,\,0\leq l\leq N-1$. In particular, $D_{\nu}(0)=C^{0}({\mathbb{R}}^{1}),\,D_{\nu}(1)=\{h\in C^{1}({\mathbb{R}}^{1}):h^{\prime}(0)=0\}$. In addition, ${\mathcal{P}}_{2\lfloor n/2\rfloor,1,e}\subseteq D_{\nu}(N)$ and $E_{1,1,e}\subseteq D_{\nu}(N)$.

If $P\in\left({\mathcal{P}}_{n,1}\cap D_{\nu}(N)\right)\setminus\{0\}$, then $P^{*}(t):=(P(t)+P(-t))/2$ belongs to ${\mathcal{P}}_{2\lfloor n/2\rfloor,1,e}\setminus\{0\}$ and for $p\in[1,{\infty})$,

[TABLE]

Similarly, if $f\in\left(E_{1,1}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})\cap D_{\nu}(N)\right)\setminus\{0\}$, then $f^{*}(t):=(f(t)+f(-t))/2$ belongs to $\left(E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})\right)\setminus\{0\}$ and for $p\in[1,{\infty})$.

[TABLE]

Therefore, it is possible to replace ${\mathcal{P}}_{2\lfloor n/2\rfloor,1,e}$ by ${\mathcal{P}}_{n,1}\cap D_{\nu}(N)$ and replace $E_{1,1,e}$ by $E_{1,1}\cap D_{\nu}(N)$ in (4.1) for $p\in[1,{\infty})$.

Theorem 4.3**.**

If $\nu\geq-1/2,\,N\in{\mathbb{Z}}^{1}_{+},$ and $p\in(0,{\infty}]$, then the limit relation

[TABLE]

is valid. In addition, there exists a function $f_{0}\in\left(E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})\right)\setminus\{0\}$ such that

[TABLE]

Note that for $N=0$ and $\nu=-1/2$ relation (4.3) was proved in [25, p. 246] by a different method, while for $\nu=-1/2$ Theorem 4.3 in more general settings was proved in [20, Theorem 1.5].

The proofs of Theorems 4.1 and 4.3 are presented in Section 6.

We also define the following sharp constant which is similar to ${\mathcal{C}}_{a}$:

[TABLE]

Certainly ${\mathcal{C}}_{a}\leq{\mathcal{C}}$ but in some cases ${\mathcal{C}}_{a}={\mathcal{C}}$.

Next, we discuss limit relations between sharp constants in multivariate inequalities of different metrics.

Corollary 4.4**.**

If $m\in{\mathbb{N}},\,N\in{\mathbb{Z}}^{1}_{+}$, and $p\in[1,{\infty}]$, then

[TABLE]

Proof.

The first relation in (4.4) follows from Corollaries 3.9 and 3.11 and Theorem 4.1 for $\nu=m/2-1$. Next, let $f\in E_{1,m}\cap L_{p}({\mathbb{R}}^{m}),\,p\in[1,{\infty}]$. Then inequality (3.7) holds for ${\sigma}=1$; therefore, $f$ has exponential type $1$ by the definition in [29, Sect. 3.1]. In addition, $\left(\Delta^{N}\right)(f)$ has exponential type $1$ as well (see [29, Sect. 3.1] or [17, Lemma 2.1 (d)]), and also $\left(\Delta^{N}\right)(f)\in L_{p}({\mathbb{R}}^{m})$ by Bernstein’s inequality (see, e. g., [29, Eq. 3.2.2(8)]).

If $p\in[1,{\infty})$, then $\lim_{|x|\to{\infty}}\left(\Delta^{N}\right)(f)(x)=0$ by [29, Theorem 3.2.5]. Therefore, there exists $x_{0}\in{\mathbb{R}}^{m}$ such that $\left\|\left(\Delta^{N}\right)(f)\right\|_{L_{\infty}({\mathbb{R}}^{m})}=\left|\left(\Delta^{N}\right)(f)(x_{0})\right|$. Setting now $f^{*}(x):=f(x_{0}-x)$, we see that $\left\|\left(\Delta^{N}\right)(f)\right\|_{L_{\infty}({\mathbb{R}}^{m})}=\left|\left(\Delta^{N}\right)(f^{*})(0)\right|$ and $\left\|f\right\|_{L_{p}({\mathbb{R}}^{m})}=\left\|f^{*}\right\|_{L_{p}({\mathbb{R}}^{m})}$. Thus ${\mathcal{C}}\leq{\mathcal{C}}_{0}$ for $p\in[1,{\infty})$.

If $p={\infty}$, then for any ${\varepsilon}>0$ there exists $x_{0}=x_{0}({\varepsilon})\in{\mathbb{R}}^{m}$ such that $\left\|\left(\Delta^{N}\right)(f)\right\|_{L_{\infty}({\mathbb{R}}^{m})}<\left|\left(\Delta^{N}\right)(f)(x_{0})\right|+{\varepsilon}\left\|f\right\|_{L_{\infty}({\mathbb{R}}^{m})}$. Setting again $f^{*}(x):=f(a-x)$, we see that $\left\|\left(\Delta^{N}\right)(f)\right\|_{L_{\infty}({\mathbb{R}}^{m})}<\left|\left(\Delta^{N}\right)(f^{*})(0)\right|+{\varepsilon}\left\|f\right\|_{L_{\infty}({\mathbb{R}}^{m})}$ and $\left\|f\right\|_{L_{\infty}({\mathbb{R}}^{m})}=\left\|f^{*}\right\|_{L_{\infty}({\mathbb{R}}^{m})}$. Thus ${\mathcal{C}}\leq{\mathcal{C}}_{0}+{\varepsilon}$ for $p={\infty}$, and the second equality in (4.4) is established. $\Box$

Corollary 4.5**.**

If $m\in\{2,\,3,\,\ldots\},\,N\in{\mathbb{Z}}^{1}_{+},\,p\in[1,{\infty}],$ and $a\in S^{m-1}$, then

[TABLE]

Proof.

The third relation in (4.5) is proved in the proof of Corollary 4.4 and the first one can be proved similarly. Finally, the second equality follows from Corollaries 3.11 and 3.13 and Theorem 4.3 for $\nu=(m-3)/2$. $\Box$

Note that the following special case of (4.5) for $N=0$,

[TABLE]

was proved in [9, Theorem 1.1 (i)] by a different method. The authors of [9] state that their proof of Theorem 1.1 is fairly nontrivial compared with [21, 25, 26, 20]. In this paper we show that an approach to limit relations between sharp constants developed in [20] can be applied to even more general relations than those in [9]. A general approach to these problems is developed in [18].

Finally, we discuss an asymptotic relation between sharp constants in the classical univariate Nikolskii-type inequality.

Corollary 4.6**.**

If $p\in[1,{\infty})$, then

[TABLE]

Proof.

We first note that

[TABLE]

This equality follows from Theorem 2.1 since conditions (C1) through (C5) are obviously satisfied for $m=1,\,B=E_{1,1}\cap L_{p,|t|}({\mathbb{R}}^{1}),\,G_{1}=\{-e^{*},e^{*}\},\,D=I,$ and $F(\Omega_{1})=L_{p,|t|}({\mathbb{R}}^{1})$, where $e^{*}$ is the identity transformation on ${\mathbb{R}}^{1}$. In other words, (4.8) follows from the following simple fact: if $f\in E_{1,1}\cap L_{p,|t|}({\mathbb{R}}^{1})$, then $(f(\cdot)+f(-\cdot))/2\in E_{1,1}\cap L_{p,|t|}({\mathbb{R}}^{1})$.

Next, Arestov and Deikalova [5, Theorem 1] proved that

[TABLE]

Then (4.7) follows from equalities (4.8) and (4.9) and relation (4.3) for $N=0$ and $\nu=0$. $\Box$

Note that a different asymptotic relation for ${\mathcal{C}}\left({\mathcal{P}}_{n,1},\,L_{p}([-1,1]),\,I\right),\,p\in(0,{\infty}],$ was proved in [16, Theorem 1.4] (see also [16, p. 94]).

5. Properties of Entire Functions and

Polynomials

Throughout the section we use the notation $D_{1}(f)(t):=f^{\prime}(t)/t$. In this section we discuss certain properties of univariate entire functions of exponential type and polynomials that are needed for the proofs of Theorems 4.1 and 4.3. We start with estimates of the error of polynomial approximation for functions from $E_{1,1,e}$ and Bernstein- and Nikolskii-type inequalities.

Lemma 5.1**.**

For $\tau\in(0,1)$ and any function $f\in E_{1,1,e}\cap L_{\infty}({\mathbb{R}}^{1}),$ there is a sequence of polynomials $P_{k}\in{\mathcal{P}}_{2\lfloor k/2\rfloor,1,e},\,k\in{\mathbb{N}}$, such that

[TABLE]

where $C_{1}:=(2/3)(1-\tau)^{3/2}$.

This result was proved by Bernstein [7] (see also [36, Sect. 5.4.4] and [1, Appendix, Sect. 83]). More precise and more general inequalities were obtained by the author in [13] and [14].

Lemma 5.2**.**

For $\tau\in(0,1)$ and any $f\in E_{1,1,e}\cap L_{\infty}({\mathbb{R}}^{1})$, there is a sequence of polynomials $P_{n}\in{\mathcal{P}}_{2\lfloor n/2\rfloor,1,e},n\in{\mathbb{N}}$, such that for $l\in{\mathbb{Z}}_{+}^{1}$, $r\in(0,{\infty}]$, ${\alpha}>-1$, and ${\beta}>-1$,

[TABLE]

Proof.

First of all, for $P_{k}\in{\mathcal{P}}_{2\lfloor k/2\rfloor,1,e},\,k\in{\mathbb{N}}$, and $l\in{\mathbb{Z}}_{+}^{1}$ we need the following crude Markov-type inequalities:

[TABLE]

Inequality (5.4) follows from A. A. Markov’s inequality [36, Sect. 4.8.7], while (5.5) is a consequence of the Mean Value Theorem and (5.4). Combining (5.4) with (5.5), we obtain (5.6) since $Be_{\nu}(f)(t)=f^{\prime\prime}(t)+(2\nu+1)D_{1}(f)(t)$.

Next, let $\{P_{k}\}_{k=1}^{\infty}$ be the sequence of polynomials from Lemma 5.1. Then using (5.6) and estimate (5.1), we obtain

[TABLE]

Hence for $n\in{\mathbb{N}},\,l\in{\mathbb{Z}}_{+}^{1},\,r\in(0,{\infty}],\,{\alpha}>-1$, and ${\beta}>-1$, we have

[TABLE]

Thus (5.3) is established. Relation (5.2) can be proved similarly if we use Lemma 5.1 and inequality (5.4). $\Box$

Lemma 5.3**.**

(a) If $l\in{\mathbb{Z}}_{+}^{1}$, and $f\in E_{{\sigma},1,e}\cap L_{{\infty}}({\mathbb{R}}^{1})$, then the following Bernstein-type inequalities hold:

[TABLE]

(b) Let ${\alpha}\geq 0$ and $p\in(0,{\infty})$. If $f\in E_{1,1,e}\cap L_{p,|t|^{{\alpha}}}({\mathbb{R}}^{1})$, then the following Nikolskii-type inequality holds:

[TABLE]

Proof.

(a) The first inequality in (5.7) is a classical Bernstein inequality [36, Sect. 4.8.2] and the second one immediately follows from the Mean Value Theorem and the first one.

(b) Inequality (5.8) for $p\in[1,{\infty})$ and ${\alpha}\geq 0$ was proved by Platonov [30, Theorem 3.5], while for $p\in(0,{\infty})$ and ${\alpha}=0$, (5.8) follows from Nikolskii’s inequality [29, Theorem 2.3.5]. If $p\in(0,1)$ and $f\in E_{1,1,e}\cap L_{p,|t|^{{\alpha}}}({\mathbb{R}}^{1})$, then $f\in L_{p}({\mathbb{R}}^{1})$; hence $f\in L_{{\infty}}({\mathbb{R}}^{1})$ by Nikolskii’s inequality. Then $f\in L_{1,|t|^{{\alpha}}}({\mathbb{R}}^{1})$ since

[TABLE]

Using estimate (5.9) and Platonov’s inequality (5.8) for $p=1$, we obtain

[TABLE]

Therefore, (5.8) for $p\in(0,1)$ follows from (5.10) with $C(p,{\alpha})\leq(C(1,{\alpha}))^{1/p}$. $\Box$

In addition to a compactness theorem for entire functions of exponential type from Proposition 3.1, we need a different type of a compactness theorem.

Lemma 5.4**.**

Let ${\mathcal{E}}_{1}$ be the set of all univariate entire functions $f(z)=\sum_{k=0}^{\infty}c_{k}z^{2k}$, satisfying the following condition: for any ${\delta}>0$ there exists a constant $C({\delta})$, independent of $f$ and $k$, such that

[TABLE]

Then for any sequence $\{f_{n}\}_{n=1}^{\infty}\subseteq{\mathcal{E}}_{1}$ there exist a subsequence $\{f_{n_{s}}\}_{s=1}^{\infty}$ and a function $f_{0}\in E_{1,1,e}$ such that for every $l\in{\mathbb{Z}}_{+}^{1}$,

[TABLE]

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

Proof.

The existence of a subsequence $\{f_{n_{s}}(z)=\sum_{k=0}^{\infty}c_{k,n_{s}}z^{2k}\}_{s=1}^{\infty}$ and a function $f_{0}(z)=\sum_{k=0}^{\infty}c_{k,0}z^{2k}$ such that for every $l\in{\mathbb{Z}}_{+}^{1},\,\lim_{s\to{\infty}}f_{n_{s}}^{(l)}=f_{0}^{(l)}$ uniformly on the disk $d_{R}:=\{z\in{\mathcal{C}}:|z|\leq R\},\,R>0$, was proved in [16, Lemma 2.6 (a)]. In particular,

[TABLE]

Next, it is easy to prove by induction in $l$ that if $f(z)=\sum_{p=0}^{\infty}c_{p}\,z^{2p}$, then

[TABLE]

Then we obtain from (5.14) for $M\in{\mathbb{N}}$

[TABLE]

where by (5.11) for ${\delta}=1$,

[TABLE]

Further, given ${\varepsilon}>0$ and $R>0$, we can choose $M=M({\varepsilon},R)$ such that $S_{2}<{\varepsilon}/2$. Finally, by (5.13), we can choose $s_{0}=s_{0}({\varepsilon},R)\in{\mathbb{N}}$ such that $S_{1}<{\varepsilon}/2$ for all $s\geq s_{0}$. Thus the second relation in (5.12) holds uniformly on $d_{R}$ as well. $\Box$

Certain inequalities of different weighted metrics for univariate polynomials are discussed in the following lemma.

Lemma 5.5**.**

For $P\in{\mathcal{P}}_{n,1},\,k\in{\mathbb{Z}}^{1}_{+},\,{\varepsilon}\in(0,1/2),\,p\in(0,{\infty}),\,{\alpha}\geq 0,$ and ${\beta}>-1,$ the following inequalities hold:

[TABLE]

Proof.

Inequality (5.15) for ${\alpha}={\beta}=0$ and $p\in(0,{\infty})$ was proved in [16, Eq. (2.10)] by using an extension of Bari’s inequality [6, Theorem 6] to $p\in(0,{\infty})$ (see [16, Lemma 2.4]). Then (5.15) for ${\alpha}={\beta}=0$ implies the estimate

[TABLE]

Therefore, (5.15) follows from (5.17) and the inequalities

[TABLE]

where $C_{6}$ and $C_{7}$ in (5.18) are independent of $P$ and $n$. To prove the first inequality in (5.18), we observe that

[TABLE]

where $C\in(0,1/3)$ is a fixed number. Next, we note that $0<C/n<1/3$, so by (5.17),

[TABLE]

Choosing now $C:=\min\{1/3,C_{5}^{-p}2^{-p-1}\}$, we obtain from (5.20)

[TABLE]

Finally, combining (5.19) and (5), we arrive at the first inequality in (5.18). Thus (5.15) is established. To prove (5.16), we use the estimate

[TABLE]

(see [36, Eq. 4.8(49)]) and inequality (5.15). $\Box$

In the next lemma, in particular, we discuss a relation between the Bessel and Gegenbauer operators.

Lemma 5.6**.**

(a) If $b\neq 0$, then

[TABLE]

(b) If $R_{n}\in{\mathcal{P}}_{n,1},\,b\neq 0,\,\nu\geq-1/2$, and $N\in{\mathbb{Z}}^{1}_{+}$, then

[TABLE]

where

[TABLE]

and

[TABLE]

(c) Let $P_{2n}=P_{2n,b(n)}$ be defined by (5.25), where $R_{n}\in{\mathcal{P}}_{n,1}$ and $b=b(n)$ satisfies the condition $\lim_{n\to{\infty}}b(n)={\infty}$. Next, let there exist a sequence of natural numbers $\{n_{s}\}_{s=1}^{\infty}$ and an even entire function $f$ such that for every $l\in{\mathbb{Z}}_{+}^{1}$,

[TABLE]

uniformly on each compact subset of ${\mathbb{R}}^{1}$. Then the following relation holds for each $t\in{\mathbb{R}}^{1}$:

[TABLE]

Proof.

(a) It suffices to prove that ${\mathcal{P}}_{2n,1,e}\subseteq S$. Let $P_{2n}\in{\mathcal{P}}_{2n,1,e}$ and let $P_{2n}(t)=V_{n}(t^{2})$, where $V_{n}\in{\mathcal{P}}_{n,1}$. Then by Taylor’s formula, $P_{2n}(t)=R_{n}\left(1-2b^{-2}t^{2}\right)$, where

[TABLE]

Therefore, $P_{2n}\in S$ and (5.22) is established.

(b) Setting $D^{*}(f)(y):=A^{*}(y)f^{\prime\prime}(y)+B^{*}(y)f^{\prime}(y)$, we see that for $y={\varphi}(t)$,

[TABLE]

where $H(t):=f({\varphi}(t))$. Then choosing

[TABLE]

we obtain (5.23) for $N=1$ from (5.28) by a straightforward calculation. Next, it follows from (5.23) for $N=1$ that for $k\in{\mathbb{N}}$

[TABLE]

Hence identity (5.23) can be proved by induction in $k$.

(c) By (5.23) and (5.24),

[TABLE]

where for any $A>0,\,U(g)(t):=t^{2}g^{\prime\prime}(t)+tg^{\prime}(t)$ is a continuous differential operator in the $L_{\infty}([-A,A])$-metric on the set of all even entire functions. Then using (5.26) and (5.29), we see that

[TABLE]

uniformly on each interval $[-A,A],\,A>0$. Thus (5.27) follows from (5.23) and (5.30). $\Box$

6. Proofs of Theorems 4.1 and 4.3

Throughout the section we use the notation $\tilde{p}=\min\{1,p\}$ for $p>0$ introduced in Section 3 and also use the operator $D_{1}(f)(t)=f^{\prime}(t)/t$ introduced in Section 5.

Proof of Theorem 4.1. We first prove the inequality

[TABLE]

Let $f$ be any function from $E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1}),\,p\in(0,{\infty}],\,\nu\geq-1/2$, and let $\tau\in(0,1)$ be a fixed number. Then using even polynomials $P_{n}\in{\mathcal{P}}_{2\lfloor n/2\rfloor,1,e}$ from Lemma 5.2 for $r={\infty}$, we obtain by (5.3) and by definition (2.1) of ${\mathcal{C}}_{0}$,

[TABLE]

Next, note that $f\in L_{\infty}({\mathbb{R}}^{1})$, by Nikolskii-type inequality (5.8). Further, applying ”triangle” inequality (3.4) and using again relation (5.3) of Lemma 5.2 for ${\alpha}=2\nu+1,\,{\beta}=0,\,l=0$, and $r=p,\,p\in(0,{\infty}]$, we have

[TABLE]

Combining (6) with (6), we obtain

[TABLE]

Letting $\tau\to 1-$ in (6), we arrive at (6) for $\nu\geq-1/2,\,N\in{\mathbb{Z}}_{+}^{1}$, and $p\in(0,{\infty}]$.

Further, we prove the inequality

[TABLE]

by constructing a nontrivial function $f_{0}\in E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})$ such that

[TABLE]

Then inequalities (6) and (6) imply (4.1). In addition, $f_{0}$ is an extremal function in (4.1), that is, relation (4.1) is valid.

It remains to construct a nontrivial function $f_{0}$, satisfying (6). We first note that

[TABLE]

This inequality follows immediately from (6). Let $P_{n}\in{\mathcal{P}}_{2\lfloor n/2\rfloor,1,e}$ be an even polynomial, satisfying the equality

[TABLE]

The existence of an extremal polynomial $P_{n}$ in (6.8) can be proved by the standard compactness argument (cf. [20]). Next, setting $Q_{n}(x):=P_{n}(x/n)$, we have from (6.8) that

[TABLE]

We can assume that

[TABLE]

Then it follows from (6), (6.10), and (6.7) that for $n\geq 2N+2$,

[TABLE]

Further, $Q_{n}\in{\mathcal{P}}_{n,1}$ and it follows from inequality (5.16) of Lemma 5.5 for ${\alpha}=2\nu+1$ and ${\beta}=0$ and from (6.11) that for any ${\varepsilon}\in(0,1/2)$ and any $k\in{\mathbb{Z}}_{+}^{1}$,

[TABLE]

Let $\{n_{r}\}_{r=1}^{\infty}$ be a subsequence of natural numbers such that

[TABLE]

Inequality (6.12) shows that the polynomial sequence $\{Q_{n_{r}}\}_{r=1}^{\infty}\subseteq{\mathcal{E}}_{1}$ satisfies the conditions of Lemma 5.4. Therefore, there exist a function $f_{0}\in E_{1,1,e}$ and a subsequence $\{Q_{n_{r_{s}}}\}_{s=1}^{\infty}$ such that

[TABLE]

uniformly on any interval $[-A,A],\,A>0$. Moreover, by (6.14) for $l=2N$ and (6.10),

[TABLE]

In addition, applying ”triangle” inequality (3.4) and using (6.14) for $l=0$, (6), and (6.10), we obtain for any interval $[-A,A],\,A>0$,

[TABLE]

Next using (6) and (6.7), we see that

[TABLE]

Therefore, $f_{0}$ is a nontrivial function from $E_{1,1.e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})$, by (6.15) and (6.17). Thus for any interval $[-A,A],\,A>0$, we obtain from (6), (6.14), and (6.15)

[TABLE]

Finally, letting $A\to{\infty}$ in (6), we arrive at (6) for $\nu\geq-1/2,\,N\in{\mathbb{Z}}_{+}^{1}$, and $p\in(0,{\infty}]$. $\Box$

Proof of Theorem 4.3. The proof is similar to the proof of Theorem 4.1 but it needs more technical details. We first prove the inequality

[TABLE]

Let $f$ be any function from $E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1}),\,p\in(0,{\infty}]$, and let $\tau\in(0,1)$ be a fixed number. It follows from Nikolskii-type inequality (5.8) that $f\in L_{\infty}({\mathbb{R}}^{1})$. Given ${\varepsilon}\in(0,1/2)$, we define

[TABLE]

where $d:=\lfloor(2\nu+2)/p\rfloor+1$. It is easy to see that $F\in E_{1,1,e}\cap L_{\infty}({\mathbb{R}}^{1})$ and

[TABLE]

In addition, we prove the equality

[TABLE]

where $\eta_{\nu,N}({\varepsilon})=0$ for $N=0$ or $\nu\geq 0$ and $|\eta_{\nu,N}({\varepsilon})|\leq C_{9}{\varepsilon}\|f\|_{L_{{\infty}}({\mathbb{R}}^{1})}$ for $N\geq 1$ and $\nu\in[-1/2,0)$. Here, $C_{9}$ is independent of ${\varepsilon}$ and $f$, and

[TABLE]

Equality (6.22) holds trivially for $N=0$ or $\nu\geq 0$. To prove (6.22) for $N\geq 1$ and $\nu\in[-1/2,0)$, we first need the Leibniz-type rule for the Bessel operator. Note that the Leibniz rule holds for the operator $D_{1}$, that is,

[TABLE]

Taking also into account the Leibniz rule for the derivative $D$, we arrive at the following formula:

[TABLE]

Next, for $\nu\in[-1/2,0),\,F(t)=f_{\varepsilon}(t)g_{\varepsilon}(t)$ by (6.20), where $f_{\varepsilon}\in E_{1-{\varepsilon},1,e}\cap L_{\infty}({\mathbb{R}}^{1})$ and $g_{\varepsilon}\in E_{{\varepsilon},1,e}\cap L_{\infty}({\mathbb{R}}^{1})$. In addition,

[TABLE]

To estimate

[TABLE]

we use identity (6) for ${\varphi}=f_{\varepsilon}$ and $\psi=g_{\varepsilon}-1$. Then the uniform norm on $[-1,1]$ of all terms in (6) with $m_{1}=\ldots=m_{N}=0$ and $s_{1}=\ldots=s_{N}=0$ can be estimated by Bernstein-type inequalities (5.7) and by the relations

[TABLE]

All those estimates do not exceed $C{\varepsilon}\left\|f\right\|_{L_{\infty}({\mathbb{R}}^{1})}$. The norms of other terms in (6) do not exceed $C{\varepsilon}\left\|f\right\|_{L_{\infty}({\mathbb{R}}^{1})}$ by Lemma 5.3 (a) as well. Combining these estimates, we obtain

[TABLE]

Therefore, setting $t=0$ in (6.25), we obtain (6.22) for $N\geq 1$ and $\nu\in[-1/2,0)$ from (6.25) and (6.26).

Next, we use even polynomials $P_{2n}\in{\mathcal{P}}_{2n,1,e}$ from Lemma 5.2 such that for every $l\in{\mathbb{Z}}_{+}^{1},$

[TABLE]

uniformly on each compact subset of ${\mathbb{R}}^{1}$. In addition, relation (5.2) of Lemma 5.2 for $N=0,\,r=p,\,{\alpha}=2\nu+1,$ and ${\beta}=\nu$ shows that

[TABLE]

By Lemma 5.6 (a), there exists $R_{n}\in{\mathcal{P}}_{n,1}$ such that for $b(n)=2\tau n$,

[TABLE]

Then relations (6.27) show that we can use Lemma 5.6 (c) for $n_{s}=s,\,s\in{\mathbb{N}}$. Therefore, we obtain by (5.27)

[TABLE]

Further, using the substitution $u=1-2(2\tau n)^{-2}t^{2}$, ”triangle” inequality (3.4), and relation (6.28), we obtain for $p\in(0,{\infty}]$ and $\nu\in[-1/2,{\infty})$

[TABLE]

Next, we prove the estimate

[TABLE]

It suffices to prove this inequality for $p\in(0,{\infty})$. For $\nu\geq 0$ inequality (6.31) follows immediately from (6.21). If $\nu\in[-1/2,0)$, then for any ${\delta}\in(0,1)$ and $p\in(0,{\infty})$,

[TABLE]

where by (6.21),

[TABLE]

and by (6.20),

[TABLE]

since $d\,p>2\nu+2$.

Collecting relations (6.32), (6.33), and (6.34), we obtain

[TABLE]

Letting ${\delta}\to 0+$ in (6) we arrive at (6.31).

Combining (6.22) and (6) with (6) and (6.31), we obtain for $p\in(0,{\infty}]$ and $\nu\in[-1/2,{\infty})$

[TABLE]

Letting $\tau\to 1-$ and ${\varepsilon}\to 0+$ in (6), we obtain

[TABLE]

Hence we arrive at (6) for $p\in(0,{\infty}]$ and $\nu\in[-1/2,{\infty})$.

Further, we prove the inequality

[TABLE]

by constructing a nontrivial function $f_{0}\in E_{1,1,e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})$ such that

[TABLE]

Then inequalities (6) and (6) imply (4.3). In addition, $f_{0}$ is an extremal function in (4.3), that is, relation (4.3) is valid.

It remains to construct a nontrivial function $f_{0}$, satisfying (6). We first note that

[TABLE]

This inequality follows immediately from (6). Let $R_{n}\in{\mathcal{P}}_{n,1}$ be a polynomial, satisfying the equality

[TABLE]

The existence of an extremal polynomial $R_{n}$ in (6.40) can be proved by the standard compactness argument. We can assume that

[TABLE]

Next, setting $Q_{2n}(t):=R_{n}(1-2(2n)^{-2}t^{2})$, we have from (6.40), (6.41), and (6.39) that for $n\geq 2N+2$,

[TABLE]

Further, $Q_{2n}\in{\mathcal{P}}_{2n,1}$, and combining inequality (5.16) of Lemma 5.5 for ${\alpha}=2\nu+1$ and ${\beta}=\nu$ with (6.41) we obtain for any ${\varepsilon}\in(0,1/2)$ and any $k\in{\mathbb{Z}}_{+}^{1}$,

[TABLE]

Let $\{n_{r}\}_{r=1}^{\infty}$ be a subsequence of natural numbers such that

[TABLE]

Inequality (6.43) shows that the polynomial sequence $\{Q_{2n_{r}}\}_{r=1}^{\infty}\subseteq{\mathcal{E}}_{1}$ satisfies the conditions of Lemma 5.4. Therefore, there exist a function $f_{0}\in E_{1,1,e}$ and a subsequence $\{Q_{2n_{r_{s}}}\}_{s=1}^{\infty}$ such that for all $l\in{\mathbb{Z}}_{+}^{1}$,

[TABLE]

uniformly on any interval $[-A,A],\,A>0$. In addition, it follows from (6.45) for $l=0$ that for $p\in(0,{\infty}],$ and $\nu\in[-1/2,{\infty})$,

[TABLE]

Then relations (6.45) show that we can use Lemma 5.6 (c) for $n_{s}=n_{r_{s}},\,s\in{\mathbb{N}}$. Therefore, we obtain by (5.27)

[TABLE]

It follows from (6.41) and (6.47) that

[TABLE]

In addition, applying ”triangle” inequality (3.4) and using (6) and (6), we obtain for any interval $[-A,A],\,A>0$,

[TABLE]

Therefore, $f_{0}$ is a nontrivial function from $E_{1,1.e}\cap L_{p,|t|^{2\nu+1}}({\mathbb{R}}^{1})$, by (6) and (6.48). Thus for any interval $[-A,A],\,A>0$, we obtain from (6), (6), (6), and (6.48)

[TABLE]

Finally, letting $A\to{\infty}$ in (6), we arrive at (6) for $\nu\geq-1/2,\,N\in{\mathbb{Z}}_{+}^{1}$, and $p\in(0,{\infty}]$. $\Box$

Acknowledgement. We are grateful to both anonymous referees for valuable suggestions.