scholarly journals A variation of functional equations related to sine type functions

2021 ◽  
Author(s):  
Bin He ◽  
Guangsheng Wei
Keyword(s):  
Functional Equation ◽  
Exponential Type ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Lagrange Interpolation ◽  
Entire Functions ◽  
Functions Of Exponential Type ◽  
Uniqueness Of The Solution ◽  
Variation Of Functional ◽  
Interpolation Nodes

In this paper, we consider a class of functional equation Q(λ)Y (λ) −P(λ)Z(λ) = η related to sine type functions, where the known P,Q are appropriate entire functions of exponential type. We are concerned with the existence and uniqueness of the solution (Y,Z) under certain circumstances. Furthermore, we modify the Lagrange interpolation to deal with the situation of the interpolation nodes being counted by multiplicities, which is significant to solve the above functional equation.

scholarly journals Asymptotics of Polynomial Interpolation and the Bernstein Constants

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Michael Revers
Keyword(s):  
Exponential Type ◽  
Chebyshev Polynomials ◽  
Lagrange Interpolation ◽  
Entire Functions ◽  
Polynomial Interpolation ◽  
Asymptotic Bounds ◽  
Functions Of Exponential Type ◽  
Chebyshev Interpolation ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials

AbstractIt is well known that the interpolation error for $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in $$L_{\infty }\left[ -1,1\right] $$ L ∞ - 1 , 1 by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when $$\alpha $$ α tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in the $$L_{\infty }$$ L ∞ norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

scholarly journals Existence and uniqueness of the solution for Lobacevsky’s functional equation

2002 ◽  
pp. 66-70
Author(s):  
Nicolae Neamtu
Keyword(s):  
Functional Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solution ◽  
Uniqueness Of The Solution

The purpose of this paper is to give a theorem for the existence and uniqueness of solution of Lobacevsky's functional equation and to effective find it.

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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