scholarly journals Nikolskii-type inequalities for trigonometric polynomials and entire functions of exponential type

1978 ◽  
Vol 25 (1) ◽  
pp. 7-18 ◽  
Author(s):  
R. J. Nessel ◽  
G. Wilmes
Keyword(s):  
Fourier Transform ◽  
Exponential Type ◽  
Compact Support ◽  
Entire Functions ◽  
Symmetric Body ◽  
Trigonometric Polynomials ◽  
Several Variables ◽  
Integrable Functions ◽  
Functions Of Exponential Type

AbstractNikolskii-type inequalities, thus inequalities between different metrics of a function, are established for trigonometric polynomials and pth power integrable functions, 0<p<∞, of several variables having Fourier transform with compact support. It is shown that certain gaps in the spectra of the functions involved may be taken into account. As an immediate consequence it follows that the general results cover the classical inequalities which are concerned with functions of rectangular type. But at the same time one may give applications to functions of type K where K is a symmetric body in Euclidean n–space.

scholarly journals Approximation characteristics of the isotropic Nikol'skii-Besov functional classes

2021 ◽  
Vol 13 (3) ◽  
pp. 851-861
Author(s):  
S.Ya. Yanchenko ◽  
O.Ya. Radchenko
Keyword(s):  
Exponential Type ◽  
Lebesgue Space ◽  
Entire Functions ◽  
Exact Order ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Several Variables ◽  
Functions Of Exponential Type ◽  
Functional Classes ◽  
Mixed Smoothness

In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,\theta}(\mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,\theta}B(\mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,\theta}(\mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(\mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 \leqslant p \leqslant q \leqslant \infty$, $(p,q)\neq \{(1,1), (\infty, \infty)\}$, $d\geq 1$. In the case $2<p=q<\infty$, $d=1$, the established estimate is also new for the classes $S^{r}_{p,\theta}B(\mathbb{R})$.

Approximation of locally integrable functions on the real line

2014 ◽  
Vol 12 (05) ◽  
pp. 1461002
Author(s):  
Xirong Chang
Keyword(s):  
Exponential Type ◽  
Real Line ◽  
Entire Functions ◽  
The Real ◽  
Integrable Functions ◽  
Functions Of Exponential Type

The aim of this paper is to extend (ψ, β)-derivatives to [Formula: see text]-derivatives for locally integrable functions on the real line and then investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type.

A weighted version of the Paley–Wiener theorem

1989 ◽  
Vol 105 (2) ◽  
pp. 389-395 ◽  
Author(s):  
T. G. Genchev
Keyword(s):  
Fourier Transform ◽  
Exponential Type ◽  
Entire Functions ◽  
Weighted Version ◽  
Cauchy Theorem ◽  
Wiener Theorem ◽  
Functions Of Exponential Type ◽  
The Fourier Transform

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

Inequalities for Entire Functions of Exponential Type

1984 ◽  
Vol 27 (4) ◽  
pp. 463-471 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Entire Functions ◽  
Indicator Function ◽  
Trigonometric Polynomials ◽  
Bernstein’S Inequality ◽  
The Real ◽  
Functions Of Exponential Type ◽  
Image Position

AbstractBernstein's inequality says that if f is an entire function of exponential type τ which is bounded on the real axis thenGenchev has proved that if, in addition, hf (π/2) ≤0, where hf is the indicator function of f, thenUsing a method of approximation due to Lewitan, in a form given by Hörmander, we obtain, to begin, a generalization and a refinement of Genchev's result. Also, we extend to entire functions of exponential type two results first proved for polynomials by Rahman. Finally, we generalize a theorem of Boas concerning trigonometric polynomials vanishing at the origin.

scholarly journals On the converse theorem of approximation in various metrics for nonperiodic functions

2014 ◽  
Vol 95 (109) ◽  
pp. 161-171
Author(s):  
Milos Tomic
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Modulus Of Smoothness ◽  
Converse Theorem ◽  
Best Approximations ◽  
Several Variables ◽  
Functions Of Exponential Type

The modulus of smoothness in the norm of space Lq of nonperiodic functions of several variables is estimated by best approximations by entire functions of exponential type in the metric of space Lp, 1 ? p ? q < ?.

Sign in / Sign up

Export Citation Format

Share Document