Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(ℝn×nK) of general type S is of the form B(ϕ,ψ) = ∫ϕ(x)ψ(x)dμ(x), ϕ, ψ ∈ sMpMp (ℝn), where μ is a positive {M}-tempered measure, i.e., for every ∈ > 0 ∫exp[-M(∈|x|)] dμ(x) < ∞. To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Positivity , Volume 2 (4) – Oct 14, 2004

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Publisher
Springer Journals
Copyright
Copyright © 1998 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009778609687
Publisher site
See Article on Publisher Site

Abstract

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(ℝn×nK) of general type S is of the form B(ϕ,ψ) = ∫ϕ(x)ψ(x)dμ(x), ϕ, ψ ∈ sMpMp (ℝn), where μ is a positive {M}-tempered measure, i.e., for every ∈ > 0 ∫exp[-M(∈|x|)] dμ(x) < ∞. To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.

Journal

PositivitySpringer Journals

Published: Oct 14, 2004

References

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