Abstract
Fourier transforms of integrable functions defined on the whole real line \(\mathbb R\) are studied in Chap. 2. First, in Sect. 2.1, the Fourier transform is defined on the Banach space \(L_1(\mathbb R)\). The main properties of the Fourier transform are handled, such as the Fourier inversion formula and the convolution property. Then, in Sect. 2.2, the Fourier transform is introduced as a bijective mapping of the Hilbert space \(L_2(\mathbb R)\) onto itself by the theorem of Plancherel. The Hermite functions, which form an orthogonal basis of \(L_2(\mathbb R)\), are eigenfunctions of the Fourier transform. In Sect. 2.3, we present the Poisson summation formula and Shannon’s sampling theorem. Finally, two generalizations of the Fourier transform are sketched in Sect. 2.5, namely the windowed Fourier transform and the fractional Fourier transform.
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References
A. Bultheel, H. Martínez, An introduction to the fractional Fourier transform and friends. Cubo 7(2), 201–221 (2005)
A. Bultheel, H. Martínez-Sulbaran, A shattered survey of the fractional Fourier transform. Manuscript (2003)
A. Bultheel, H.E. Martínez-Sulbaran, Computation of the fractional Fourier transform. Appl. Comput. Harmon. Anal. 16(3), 182–202 (2004)
K. Chandrasenkharan, Classical Fourier Transforms (Spinger, Berlin, 1989)
C. Févotte, N. Bertin, J.L. Durrieu, Nonnegative matrix factorization with the Itakura-Saito divergence: with application to music analysis. Neural Comput. 21(3), 793–830 (2009)
D. Gabor, The theory of communication. J. IEE 93, 429–457 (1946)
C. Gasquet, P. Witomski, Fourier Analysis and Applications. Filtering, Numerical Computation, Wavelets (Springer, Berlin, 1999)
L. Grafakos, Classical Fourier Analysis, 2nd edn. (Springer, New York, 2008)
K. Gröchenig, Foundations of Time–Frequency Analysis (Birkhäuser, Boston, 2001)
J.J. Healy, M.A. Kutay, H.M. Ozaktas, J.T. Sheridan, Linear Canonical Transforms. Theory and Applications (Springer, New York, 2016)
V.A. Kotelnikov, On the transmission capacity of the “ether” and wire in electrocommunications. Translated from Russian, in Modern Sampling Theory: Mathematics and Application (Birkhäuser, Boston, 2001), pp. 27–45
N.N. Lebedev, Special Functions and Their Applications, Translated from Russian (Dover, New York, 1972)
H.Q. Nguyen, M. Unser, J.P. Ward, Generalized Poisson summation formulas for continuous functions of polynomial growth. J. Fourier Anal. Appl. 23(2), 442–461 (2017)
H. Nyquist, Certain factors affecting telegraph speed. Bell Syst. Tech. J. 3(2), 324–346 (1924)
H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, 2001)
C.E. Shannon, Communication in the presence of noise. Proc. I.R.E. 37, 10–21 (1949)
E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971)
M. Unser, Sampling – 50 years after Shannon. Proc. IEEE 88, 569–587 (2000)
E.T. Whittaker, On the functions which are represented by the expansions of the interpolation-theory. Proc. R. Soc. Edinb. 35, 181–194 (1914)
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Plonka, G., Potts, D., Steidl, G., Tasche, M. (2018). Fourier Transforms. In: Numerical Fourier Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04306-3_2
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DOI: https://doi.org/10.1007/978-3-030-04306-3_2
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