Extremal Problems for Positive-Definite Bandlimited Functions. III. The Maximum Number of Zeros in an Interval $[0,T]$

1983 ◽  
Vol 14 (2) ◽  
pp. 258-268 ◽  
Author(s):  
B. F. Logan
Keyword(s):  
Extremal Problems ◽  
Positive Definite ◽  
Bandlimited Functions ◽  
Number Of Zeros

scholarly journals APPLICATIONS OF AN INTERSECTION FORMULA TO DUAL CONES

2017 ◽  
Vol 97 (1) ◽  
pp. 94-101
Author(s):  
DÁNIEL VIROSZTEK
Keyword(s):  
Banach Spaces ◽  
Duality Theorem ◽  
Extremal Problems ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Trigonometric Polynomials ◽  
Dual Cones ◽  
Intersection Formula

We give a succinct proof of a duality theorem obtained by Révész [‘Some trigonometric extremal problems and duality’, J. Aust. Math. Soc. Ser. A 50 (1991), 384–390] which concerns extremal quantities related to trigonometric polynomials. The key tool of our new proof is an intersection formula on dual cones in real Banach spaces. We show another application of this intersection formula which is related to integral estimates of nonnegative positive-definite functions.

scholarly journals Some trigonometric extremal problems and duality

1991 ◽  
Vol 50 (3) ◽  
pp. 384-390 ◽  
Author(s):  
Szilárd GY. Révész
Keyword(s):  
Functional Analysis ◽  
Crucial Role ◽  
Duality Theorem ◽  
Minimax Theorem ◽  
Zeta Functions ◽  
Prime Number Theorem ◽  
Extremal Problems ◽  
Duality Results ◽  
Number Of Zeros

AbstractIn this paper we present a minimax theorem of infinite dimension. The result contains several earlier duality results for various trigonometrical extremal problems including a problem of Fejér. Also the present duality theorem plays a crucial role in the determination of the exact number of zeros of certain Beurling zeta functions, and hence leads to a considerable generalization of the classical Beurling's Prime Number Theorem. The proof used functional analysis.

scholarly journals Zeros of the Epstein zeta function to the right of the critical line

2020 ◽  
pp. 1-12
Author(s):  
YOUNESS LAMZOURI
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Critical Line ◽  
Quadratic Field ◽  
Positive Definite ◽  
Real Numbers ◽  
Positive Definite Quadratic Form ◽  
Number Of Zeros ◽  
Epstein Zeta Function ◽  
The Right

Abstract Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${{\mathbb{Q}}(\sqrt D)}$ . We denote by N E (σ1, σ2, T) the number of zeros of E(s, Q) in the rectangle σ1 < Re(s) ≤ σ2 and T ≤ Im (s) ≤ 2T, where 1/2 < σ1 < σ2 < 1 are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for N E (σ1, σ2, T), obtaining a saving of a power of log T in the error term.

scholarly journals On the zeros of Epstein zeta functions

2014 ◽  
Vol 26 (6) ◽  
Author(s):  
Yoonbok Lee
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Critical Line ◽  
Zeta Functions ◽  
Upper Bounds ◽  
Positive Definite ◽  
Asymptotic Formulas ◽  
Positive Definite Quadratic Form ◽  
Number Of Zeros ◽  
Existence Of Zeros

AbstractWe investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

scholarly journals On Pointwise Estimates of Positive Definite Functions With Given Support

2006 ◽  
Vol 58 (2) ◽  
pp. 401-418 ◽  
Author(s):  
Mihail N. Kolountzakis ◽  
Szilárd Gy. Révész
Keyword(s):  
Extremal Problems ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Trigonometric Polynomials ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Convex Domains ◽  
Extremal Values ◽  
Limiting Case ◽  
The Relationship

AbstractThe following problem has been suggested by Paul Turán. Let Ω be a symmetric convex body in the Euclidean space ℝd or in the torus . Then, what is the largest possible value of the integral of positive definite functions that are supported in Ω and normalized with the value 1 at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in ℝ. That is, under the same conditions and normalizations, the supremum of possible function values at z is to be found for any given point z ∈ Ω. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to ℝd and to non-convex domains as well.Here we present another approach to the problem, giving the solution in ℝd and for several cases in . Actually, we elaborate on the fact that the problemis essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for ℝd and that for is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

Sign in / Sign up

Export Citation Format

Share Document