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Applications of Fourier Transforms to Generalized Functions

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The generalized function is one of the important branches of mathematics that has enormous application in practical fields. Especially, its applications to the theory of distribution and signal processing are very much noteworthy. The method of generating solutions is the Fourier transform, which has great applications to the generalized functions. These two branches of mathematics are very important for solving practical problems. While I was at Imperial College London (1966–1969), I attended many lectures delivered on fluid mechanics topics by Sir James Lighthill, FRS. At that time I was unable to understand many of the mathematical ideas in connection with the generalized function and why we need this abstract mathematics in the applied field. I tried to follow Lighthill’s book An Introduction to Fourier Analysis and Generalized Functions, published by Cambridge University Press, 1964. His book is very compact (only 79 pages) and extremely stimulating, but he has written it so elegantly that unless one has good mathematical background, the book is very hard to follow. I understand that a non-expert reader will find the book very hard to follow because of its compactness and too many cross-references. Mathematical details are very minimal and he sequentially explains from one step to another skipping many intermediate steps by the cross references. Lighthill followed the ideas originally described by Professor George Temple’s Generalized Functions, Proc. Roy. Soc. A, 228, 175–190, (1955). Lighthill kept the theory part as described by Temple. In Dalhousie University I used to give a course on Mathematical Methods and their Applications to the undergraduate and graduate students for several years. I used Fourier transforms and generalized functions in that course. To make it understandable to the student I had to take recourse to some engineering textbooks where the applications are found in this subject. I followed some engineering application of generalized functions and its solution technique using the Fourier transform method.
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