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Uniqueness of the Continuation of a Certain Function to a Positive Definite Function
- Source :
- Mathematical Notes. 107:639-652
- Publication Year :
- 2020
- Publisher :
- Pleiades Publishing Ltd, 2020.
-
Abstract
- In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| < a, can be extended to a positive definite one on R if and only if 0 < a ≤ 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| ≤ a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a ∈ [1/2,1] and Re ϕ(x) = 1 - |x|, |x| ≤ a, for some ϕ ∈ T, then ϕ(x) = (1 - |x|) +, x G R. In addition, for any a G (0,1/2), there exists a function ϕ ∈ T such that ϕ(x) = 1 - |x|, |x| ≤ a, but ϕ(x) ≠ (1 - |x|)+. Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.
- Subjects :
- Class (set theory)
Continuous function (set theory)
General Mathematics
010102 general mathematics
02 engineering and technology
Function (mathematics)
Positive-definite matrix
Extension (predicate logic)
01 natural sciences
Combinatorics
020303 mechanical engineering & transports
0203 mechanical engineering
Positive-definite function
Interval (graph theory)
Uniqueness
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 15738876 and 00014346
- Volume :
- 107
- Database :
- OpenAIRE
- Journal :
- Mathematical Notes
- Accession number :
- edsair.doi...........783710b8bf1ca85a120245ef907767cc