A Handbook of Fourier Theorems
This book is concerned with the well-established mathematical technique known as Fourier analysis (or alternatively as harmonic or spectral analysis). It is a handbook comprising a collection of the most important theorems in Fourier analysis, presented without proof in a form that is accurate but also accessible to a reader who is not a specialist mathematician. The technique of Fourier analysis has long been of fundamental importance in the physical sciences, engineering and applied mathematics, and is today of particular importance in communications theory and signal analysis. Existing books on the subject are either rigorous treatments, intended for mathematicians, or are intended for non-mathematicians, and avoid the finer points of the theory. This book bridges the gap between the two types. The text is self-contained in that it includes examples of the use of the various theorems, and any mathematical concepts not usually included in degree courses in physical sciences and engineering are explained. This handbook will be of value to postgraduates and research workers in the physical sciences and in engineering subjects, particularly communications and electronic engineering.
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The book starts with interpretation of Lebesque integrals. Unfortunately it does not cover the abstract harmonic analysis .
The three important theorems in Fournier analysis namely Inversion,convolution and differentiation theorems are mentioned. Inversion theorem is ambiguous in that it does not state to what class of functions the f(x) of the theorem ,whether the integrals are to be interpreted as Reimans,Libesque ,generalized or some other form of the integral.Fourier symbol ʃ +/-α ƒ(x)δx
is to be interpreted as direct Lebesque integral or as some limit.
Fourriers transformation function of f(x)=1 is the Dirac Delta function δ(y). f(x)=1 is represented by limit a α of exp(-π a2y2).Delta function is an example of generalized function.
Fourrier theory is based on theory of Integration of G.F.B Reinmaan (1826-66) and an alternative approach was suggested by H L Lebesque (1875-1941) considered more simple and powerful.
Reimaann Integration:-Real function of real variables between 2 finite limits a and b .Divide this area into vertical strips .The area of each of the strip approximated to product of width of strip and sampled value of function at the same arbitrary point within the strip.Consider the limit of sum of approximate areas as width of strip tend to zero and number of such arips tends to infinity.
A Null set measures zero. An empty set = A null set
Reimaana and Lebesque has different meaning for the symbols (For Reimaan the symbols are equal by definition) .The text then continue to examine Minskouski inequality,Holder’s Theorem,Yong’s theorem ,Fubiin and Tonelli theorems .
Absoloute and uniform continuity: Differentiation and Integration are regarded as inverse.The primitive and the derivative.Is a set finite or infinite ? Close or open ? Or both ? This question is answered for Brahman and Prakruthi in mathematics of ancient India.Uniform continuity on absolute continuity form a hierarchy of conditions , each more stringent than its predecessor.This result , a generalization of result first obtained in the west ,by Reimaan in restricted context of Reimannaian integration.
The sin cosine oscillation functions as a continuous function of changing values and as harmonic waves is the ancient Nadabrahman concept.The areas of neighbouring oscillations in f(x) cos ax cancel to zero even more accurately as a α .In the astronomical texts of Varahamihira and others before him as he gives in Panchasidhanthika explains this sin cosine functions mathematically and musicological texts explains it harmonic wave patterns while Thanthric texts integrate these approaches .
This handbook for degree and postgraduate students is a basic text to understand Fourrier theory