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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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