Representation Formulas for Integrable and Entire Functions of Exponential Type II

1991 ◽  
Vol 43 (1) ◽  
pp. 34-47 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Entire Functions ◽  
Type Ii ◽  
The Real ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

We adopt the terminology and notations of [5]. If f ∈ Bτ is an entire function of exponential type τ bounded on the real axis then we have the complementary interpolation formulas [1, p. 142-143] andwhere t, γ are reals and

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.

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.

On the approximation of analytic functions in a strip

1985 ◽  
Vol 97 (3) ◽  
pp. 381-384 ◽  
Author(s):  
Dieter Klusch
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Uniform Approximation ◽  
Analytic Functions ◽  
Trigonometric Polynomials ◽  
The Real ◽  
Image Position ◽  
Class Of Functions

1. Letand denote by Aδ the class of functions f analytic in the strip Sδ = {z = x + iy| |y| < δ}, real on the real axis, and satisfying |Ref(z)| ≤ 1,z∊Sδ. Then N.I. Achieser ([1], pp. 214–219; [8], pp. 137–8, 149) proved that each f∊Aδ can be uniformly approximated on the whole real axis by an entire function fc of exponential type at most c with an errorwhere ∥·∥∞ is the sup norm on ℝ. Furthermore ([7], pp. 196–201), if f∊Aδ is 2π-periodic, then the uniform approximation Ẽn (Aδ) of the class Aδ by trigonometric polynomials of degree at most n is given by

scholarly journals Some representation formulae for entire functions of exponential type

1988 ◽  
Vol 37 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Entire Functions ◽  
Bernoulli Numbers ◽  
First Case ◽  
Functions Of Exponential Type ◽  
Explicit Formulae ◽  
Image Position ◽  
Fejér Mean ◽  
Derivatives Of

We obtain some explicit formulae for series of the typewhere f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

scholarly journals Representation formulas for entire functions of exponential type and generalized bernoulli polynomials

1998 ◽  
Vol 64 (3) ◽  
pp. 307-316 ◽  
Author(s):  
C. Frappier
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
General Class ◽  
Bessel Functions ◽  
Entire Functions ◽  
Bernoulli Polynomials ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Generalized Bernoulli Polynomials

AbstractWe introduce a sequence of polynomials which are extensions of the classic Bernoulli polynomials. This generalization is obtained by using the Bessel functions of the first kind. We use these polynomials to evaluate explicitly a general class of series containing an entire function of exponential type.

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