By R. N. Bracewell (auth.), John F. Price (eds.)

The first systematic tools of Fourier research date from the early eighteenth century with the paintings of Joseph Fourier at the challenge of the circulate of warmth. (A short historical past is inside the first paper.) Given the preliminary tempera­ ture in any respect issues of a sector, the matter used to be to figure out the alterations within the temperature distribution through the years. figuring out and predicting those alterations was once very important in such components because the dealing with of metals and the decision of geological and atmospheric temperatures. in short, Fourier spotted that the answer of the warmth diffusion challenge used to be uncomplicated if the preliminary temperature dis­ tribution was once sinusoidal. He then asserted that any distri­ bution might be decomposed right into a sum of sinusoids, those being the harmonics of the unique functionality. This intended that the final answer may possibly now be bought through summing the solu­ tions of the part sinusoidal difficulties. This striking skill of the sequence of sinusoids to explain all "reasonable" features, the sine qua non of Fourier research and synthesis, has resulted in the regimen use of the tools originating with Fourier in an excellent range of parts - astrophysics, computing, economics, electric engineering, geophysics, info concept, scientific engineering, optics, petroleum and mineral exploration, quan­ tum physics and spectroscopy, to call a few.

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Exp(-~w2) s~owed that f = 0, IF(w) I .. (t (w £ £ R) R) In 1933 Hardy [9] or f is a constant multiple of exp(-at), or there ~re infini~ely many suc2 functions f according as a~ > n , a~ = n, or a~ < n . Hence it is impossible to construct a nonzero signal f which, along with its Fourier transform is "rapidly decreasing" at infinity. 3 Entropy Considerations. Following Shannon [22], the entropy of a signal f is defined as H(f) = -f If(t) Iloglf(t) Idt , and similarly for its Fourier transform. F.

A simple estimate of this error is given by the following result. Theorem. tinuous) LetECIR be an Q-transversal set. 1) for all Proof. t in Given 1R. t in 1R and N in {1,2, ••. }, define '¥N, t (W)_; [exp(27dtw) - LIn I<;Nexp(7dnw/Q )

UNCERTAINTY PRINCIPLES AND SAMPLING THEOREMS 43 7. J. Fuchs, On the eigenvalues of an integral equation arising in the theory of band-limited signals, ~. Math. Anal. App1. 9: 317-330 (1964). 8. D. Gabor, Theory of communication, J. Inst. E1ectr. Engnrs. 93(3): 429-457 (1946). 9. H. Hardy, A theorem concerning Fourier transforms, J. London Math. Soc. 8: 227-231 (1933). 10. 1. , A note on entropy, Amer. J. Math. 11. J. Jerri, The Shannon sampling theorem - its various extensions and applications, Proc.

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