Extremal Problems for Positive-Definite Bandlimited Functions. II. Eventually Negative Functions

1983 ◽  
Vol 14 (2) ◽  
pp. 253-257 ◽  
Author(s):  
B. F. Logan
Keyword(s):  
Extremal Problems ◽  
Positive Definite ◽  
Bandlimited Functions

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 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.

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