Abstract
This rather long chapter is provided mainly for the benefi t of those who are interested in studying wave phenomena but whose mathematical background does not necessarily include modern functional analysis and operator theory. For such scientists this chapter is intended to be a guide through the material available whilst for those with more mathematical background it can act as a source. Virtually no proofs are given. Details of these can be found in the references cited. Despite the fact that much of the material might, at first sight, give the impression of being unnecessarily abstract, nevertheless, it will be seen to be of considerable use in justifying various approaches to the development of constructive methods of solving physical problems.
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References and Further Reading
N.I. Akheiser and L.M. Glazman: Theory of Linear Operators in Hilbert Space, Pitman-Longman, London, 1981.
J. Arsac: Fourier Transforms and the Theory of Distributions, Prentice Hall, New York, 1966.
J.W. Dettman: Mathematical Methods in Physics and Engineering, McGraw-Hill, New York, 1962.
G. Helmberg: Introduction to Spectral Theory in Hilbert Space, Elsevier, New York, 1969.
E. Kreyszig: Introductory Functional Analysis with Applications, Wiley, New York, 1978.
F. Riesz and B. Sz-Nagy: Functional Analysis, Ungar, New York, 1955.
G.F. Roach: Greens Functions (2nd Ed.), Cambridge Univ. Press, London, 1970/1982.
G.F. Roach: An Introduction to Linear and Nonlinear Scattering Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol 78, Longman, Essex 1995.
W. Rudin: Principles of Mathematical Analysis (3rd Ed.), McGraw-Hill, New York, 1976.
I.N. Sneddon: Fourier Transforms, McGraw-Hill, New York, 1951.
E.C. Titchmarsh: Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, 1937.
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(2008). Preliminary Mathematical Material. In: An Introduction to Echo Analysis. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84628-852-4_3
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DOI: https://doi.org/10.1007/978-1-84628-852-4_3
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