Functions of Exponential Type are Differences of Functions of Bounded Index

1977 ◽  
Vol 20 (4) ◽  
pp. 479-483 ◽  
Author(s):  
Shantilal N. Shah
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Functions Of Exponential Type ◽  
Image Position

The notion of entire function of Bounded Index is by now well established. It may be stated as follows.An entire function f(z) is said to be of Bounded Index if for some fixed sfor all n and all z. (See [1], [2].)

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.

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.

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 Weighted Markov-Bernstein inequalities for entire functions of exponential type

2014 ◽  
Vol 96 (110) ◽  
pp. 181-192 ◽  
Author(s):  
Doron Lubinsky
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Entire Functions ◽  
Functions Of Exponential Type ◽  
Bernstein Inequalities

We prove weighted Markov-Bernstein inequalities of the form ???? |f?(x)|pw(x) dx ? C(? + 1)p ???? |f(x)|pw(x) dx Here w satisfies certain doubling type properties, f is an entire function of exponential type ? ?, p > 0, and C is independent of f and ?. For example, w(x) = (1 + x2)? satisfies the conditions for any ? ? R. Classical doubling inequalities of Mastroianni and Totik inspired this result.

Remarks on Entire Functions of Exponential Type

1986 ◽  
Vol 29 (3) ◽  
pp. 365-371
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Classical Result ◽  
Entire Functions ◽  
Functions Of Exponential Type

AbstractA classical result of Laguerre says that if P is a polynomial of degree n such that P(z) ≠ 0 for | z | < 1 then (ξ - z)P' (z) + nP(z) ≠ 0 for | z | < 1 and | ξ | < 1. Rahman and Schmeisser have obtained an extension of that result to entire functions of exponential type: if f is an entire function of exponential type τ, bounded on ℝ, such that hf(π/2) = 0 then (ξ- l)f'(z) + iτ(z) ≠ 0 for Im(z) > 0 and | ξ | < 1, whenever f(z) ≠ 0 if Im(z) > 0. We obtain a new proof of that result. We also obtain a generalization, to entire functions of exponential type, of a result of Szegö according to which the inequality | P(Rz) — P(z) | < Rn - 1, | z | ≤ 1, R ≥ 1, holds for all polynomials P, of degree ≤ n, such that | P(z) | ≤ 1 for | z | ≤ 1.

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 Bounding Sn(t) on the Riemann hypothesis

2017 ◽  
Vol 164 (2) ◽  
pp. 259-283 ◽  
Author(s):  
EMANUEL CARNEIRO ◽  
ANDRÉS CHIRRE
Keyword(s):  
Exponential Type ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Final Section ◽  
Extremal Functions ◽  
Functions Of Exponential Type ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta ◽  
Image Position

AbstractLet $S(t) = {1}/{\pi} \arg \zeta \big(\hh + it \big)$ be the argument of the Riemann zeta-function at the point 1/2 + it. For n ⩾ 1 and t > 0 define its iterates $$\begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,\d\tau\, + \delta_n\,, \end{equation*}$$ where δn is a specific constant depending on n and S0(t) ≔ S(t). In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that Sn(t) = O(log t/(log log t)n + 1). The order of magnitude of this estimate was never improved up to this date. The best bounds for S(t) and S1(t) are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate $$\begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*}$$ for all n ⩾ 2, with the constants C±n decaying exponentially fast as n → ∞. This improves (for all n ⩾ 2) a result of Wakasa, who had previously obtained such bounds with constants tending to a stationary value when n → ∞. Our method uses special extremal functions of exponential type derived from the Gaussian subordination framework of Carneiro, Littmann and Vaaler for the cases when n is odd, and an optimized interpolation argument for the cases when n is even. In the final section we extend these results to a general class of L-functions.

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.

On Generalized Powers of the Difference Operator

1965 ◽  
Vol 17 ◽  
pp. 124-129
Author(s):  
K. R. Unni
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Difference Operator ◽  
Alternative Definition ◽  
Averaging Operator ◽  
The Difference ◽  
Definition Of ◽  
Image Position

Sumner (3) discussed for arbitrary real λ and h, where the averaging operator ∇h is defined by(1.1)when f(z) is an entire function of exponential type <2π/|h|. Boas (2) gave an alternative definition of ∇h which gave Sumner's results quickly and showed that his definition is equivalent to that of Sumner.

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