Fourier analysis is one of the most useful tools in many applied
sciences. The recent developments of wavelet analysis indicate that
in spite of its long history and well-established applications, the
field is still one of active research. This text bridges the gap
between engineering and mathematics, providing a rigorously
mathematical introduction of Fourier analysis, wavelet analysis and
related mathematical methods, while emphasizing their uses in
signal processing and other applications in communications
engineering. The interplay between Fourier series and Fourier
transforms is at the heart of signal processing, which is couched
most naturally in terms of the Dirac delta function and Lebesgue
integrals. The exposition is organized into four parts. The first
is a discussion of one-dimensional Fourier theory, including the
classical results on convergence and the Poisson sum formula. The
second part is devoted to the mathematical foundations of signal
processing sampling, filtering, digital signal processing. Fourier
analysis in Hilbert spaces is the focus of the third part, and the
last part provides an introduction to wavelet analysis,
time-frequency issues, and multiresolution analysis. An appendix
provides the necessary background on Lebesgue integrals.
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