scholarly journals On the final set of a real entire function of exponential type

1987 ◽  
Vol 99 (4) ◽  
pp. 700-700
Author(s):  
Murali Rao ◽  
Li-Chien Shen
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Real Entire Function

Functions in the Schwartz algebra that are invertible in the sense of Ehrenpreis

2019 ◽  
Vol 484 (1) ◽  
pp. 7-11
Author(s):  
N. F. Abuzyarova
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Zero Set ◽  
Schwartz Algebra

We consider the problem of obtaining the restrictions on the zero set of an entire function of exponential type under which this function belongs to the Schwartz algebra and invertible in the sense of Ehrenpreis.

scholarly journals Bank-Laine Functions with Real Zeros

2020 ◽  
Vol 20 (3-4) ◽  
pp. 653-665
Author(s):  
J. K. Langley
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Explicit Representation ◽  
Trigonometric Functions ◽  
Exponent Of Convergence ◽  
Real Zeros ◽  
Real Entire Function ◽  
Bounded Below ◽  
Quasiconformal Surgery

AbstractSuppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that $$E'(z) = \pm 1$$ E ′ ( z ) = ± 1 whenever $$E(z) = 0$$ E ( z ) = 0 . Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
George Csordas ◽  
Anna Vishnyakova
Keyword(s):  
Entire Function ◽  
Open Problem ◽  
A Priori ◽  
Sufficient Conditions ◽  
Class Definition ◽  
Lindelöf Theorem ◽  
Order And Type ◽  
Geometric Result ◽  
Real Entire Function ◽  
Inequality Theorem

AbstractThe principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

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.

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

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

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