Fractional Fourier transformation of Schwartz test functions
Fractional calculus involving both the fractional derivatives and the fractional integrals has been found to evaluate several mathematical problems through its elegant mathematical concepts. Besides Section 1, introducing the background of the subject by showing some researches on similar themes and Section~2, which embraces some definitions and two theorems that are used in obtaining the main results, Section~3 involves four propositions as well as some definitions and examples, and there are eight theorems that justify the intentions to study the fractional Fourier transformation on the Schwartz distribution space $\mathscr{S}$ of test functions. It is claimed that the method of obtaining results is totally different from the one used by \textit{A. C. McBride} and \textit{F. H. Kerr} [IMA J. Appl. Math. 39, No. 2, 159--175 (1987; Zbl 0644.44005)] and \textit{F. H. Kerr} [Proc. R. Soc. Edinb., Sect. A, Math. 108, No. 1--2, 133--143 (1988; Zbl 0672.42006)], in the sense that the authors of the present paper have employed the method of the paper by \textit{A. I. Zayed} [``A convolution and product theorem for the fractional Fourier transform'', IEEE Signal Process. Lett. 5(4), 101--103 (1998; \url{doi:10.1109/97.664179})], which is purely analytic, in order to extend the definition of fractional Fourier transform to the space of tempered distributions~$\mathscr{S}'$. Also investigated are the fractional Fourier transform between test functions and tempered distributions (Theorem~4) and between two tempered distributions, and that all the convolution operators involved are continuous. Moreover, the fractional Fourier transform is shown to help solving the non-homogeneous heat equation (Example~2). Theorems~7 and 8 along with some definitions and Proposition~5 explain fractional convolution operators on the Schwartz distribution space and the tempered distribution space and study their continuity. Finally, it may not be out of place to mention (and at the same time advise the authors) that the reviewer and his coauthors may have been the first ones to define fractional calculus theory for Boehmians (which have a mention in line~3 on p.~600).
scientific article; zbMATH DE number 7140952
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