Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

2014 ◽  
Vol 30 (10) ◽  
pp. 1748-1762
Author(s):  
Yong Ping Liu ◽  
Chun Yuan Song
Keyword(s):  
Exponential Type ◽  
Best Approximation ◽  
Entire Functions ◽  
Power Weight ◽  
Functions Of Exponential Type

scholarly journals Sharp estimates of approximation of classes of differentiable functions by entire functions

2021 ◽  
pp. 3
Author(s):  
V.F. Babenko ◽  
A.Yu. Gromov
Keyword(s):  
Exponential Type ◽  
Best Approximation ◽  
Entire Functions ◽  
Sharp Estimate ◽  
Sharp Estimates ◽  
The Best Approximation ◽  
Differentiable Functions ◽  
Functions Of Exponential Type

In the paper, we find the sharp estimate of the best approximation, by entire functions of exponential type not greater than $\sigma$, for functions $f(x)$ from the class $W^r H^{\omega}$ such that $\lim\limits_{x \rightarrow -\infty} f(x) = \lim\limits_{x \rightarrow \infty} f(x) = 0$,$$A_{\sigma}(W^r H^{\omega}_0)_C = \frac{1}{\sigma^{r+1}} \int\limits_0^{\pi} \Phi_{\pi, r}(t)\omega'(t/\sigma)dt$$for $\sigma > 0$, $r = 1, 2, 3, \ldots$ and concave modulus of continuity.Also, we calculate the supremum$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

scholarly journals Meansquare Approximation of Function Classes, Given on the all Real Axis R by the Entire Functions of Exponential Type

2016 ◽  
Vol 6 ◽  
pp. 1-12 ◽  
Author(s):  
Sergey B. Vakarchuk
Keyword(s):  
Exponential Type ◽  
Best Approximation ◽  
Fractional Derivatives ◽  
Entire Functions ◽  
Weak Equivalence ◽  
The Best Approximation ◽  
Function Classes ◽  
Functions Of Exponential Type ◽  
Specific Restriction ◽  
Derivatives Of

K-functionals K (f, t, L2(R), L2β(R), which defined by the fractional derivatives of order β>0, have been considered in the space L2(R). The relation K (f, tβ, L2(R), L2β(R) ≈ ωβ (f, t) (t>0) was obtained in the sense of the weak equivalence, where ωωβ (f, t) is the module of continuity of the fractional order β for a function f є L2(R). Exact values of the best approximation by entire functions of exponential type v∏, v є (0, ∞) have been computed for the classes of functions, given by the indicated K-functionals and majorants Ψ satisfying specific restriction. Kolmogorov, Bernsteinand linear mean v-widths were obtained for indicated classes of functions.

Representation Formulas for Integrable and Entire Functions of Exponential Type I

1988 ◽  
Vol 40 (04) ◽  
pp. 1010-1024 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Conjugate Function ◽  
Interpolation Formula ◽  
Type I ◽  
Additional Hypothesis ◽  
Period 2 ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

Let Bτ denote the class of entire functions of exponential type τ (>0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143] 1.1 where t, γ are real numbers and is the so called conjugate function of f. Let us put 1.2 The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε > 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.

scholarly journals Inequalities and representation formulas for functions of exponential type

1995 ◽  
Vol 58 (1) ◽  
pp. 15-26
Author(s):  
C. Frappier ◽  
P. Olivier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Representation Formula ◽  
Bernstein’S Inequality ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Bernstein's Inequality

AbstractWe generalise the classical Bernstein's inequality: . Moreover we obtain a new representation formula for entire functions of exponential type.

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