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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Fourier transforms

Author: C. Nasim
Journal: Trans. Amer. Math. Soc. 191 (1974), 45-51
MSC: Primary 44A05
MathSciNet review: 0342964
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Abstract: If $ f(x)$ and $ g(x)$ satisfy the equations

$\displaystyle g(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}g(t){k_1}(xt)dt,$

then we call f and g a pair of $ {k_1}$-transforms, where

$\displaystyle {k_1} = \frac{1}{{2\pi i}}\int _{{\raise0.5ex\hbox{$\scriptstyle ... ...r0.25ex\hbox{$\scriptstyle 2$}} + i\infty }\frac{{K(s)}}{{1 - s}}{x^{1 - s}}ds.$

In this paper alternative sets of conditions are established for f and g to be $ {k_1}$-transform provided $ K(s)$ is decomposable in a special way. These conditions involve simpler functions, which replace the kernel $ {k_1}(x)$. Results are proved for the function spaces $ {L^2}$. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.

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Keywords: Fourier transform, Fourier kernel, Mellin transform, $ {L^2}$-class, convergence in mean, the Parseval theorem, Bessel functions
Article copyright: © Copyright 1974 American Mathematical Society

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