Dual Integral Equations with Trigonometric Kernels and Tempered Distributions

Of concern here are two theorems pertaining to the questions of existence and uniqueness of solutions of the dual integral equations \[ \begin{gathered} \frac{1}{{\sqrt {2\pi } }}\int_{ - \infty }^\infty {| \xi |^\alpha \psi (\xi )e^{i\xi x} d\xi } = f(x),\quad 0 \leqq | x | < 1,\quad \alpha = \p...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 1972-08, Vol.3 (3), p.413-421
Main Author: Srivastav, R. P.
Format: Article
Language:English
Subjects:
Fourier transforms
Integral equations
Integrals
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Summary:Of concern here are two theorems pertaining to the questions of existence and uniqueness of solutions of the dual integral equations \[ \begin{gathered} \frac{1}{{\sqrt {2\pi } }}\int_{ - \infty }^\infty {| \xi |^\alpha \psi (\xi )e^{i\xi x} d\xi } = f(x),\quad 0 \leqq | x | < 1,\quad \alpha = \pm 1, \hfill \\ \frac{1}{{\sqrt {2\pi } }}\int_{ - \infty }^\infty {\psi (\xi )e^{i\xi x} d\xi } = 0,\quad | x | > 1. \hfill \\ \end{gathered} \] The integrals are interpreted as the classical Abel limits. This furnishes a computational device for the evaluation of Fourier transforms of tempered distributions. Using the theories of Fourier transforms and singular integral equations, an explicit solution is constructed, which is shown to be unique.
ISSN:0036-1410
1095-7154
DOI:10.1137/0503039