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One of my favourite books for harmonic analysis is Fourier analysis by Javier Duoandikoetxea. In fact, I would recommend that as your first port of call for learning harmonic analysis if you already have some background (such as an undergraduate/postgraduate course in harmonic analysis). That is a terrific reference for background regardless of what you want ...

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For the link to applications you might try Palle Jorgensen´s: Analysis and Probability. Wavelets, Signals, Fractals. As a special feature, this book contains a dictionary of the use of technical terms in different disciplines.

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I know that the discrete Mellin transform was defined by V.S.Ryko: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=5138&option_lang=rus English reference: Soviet Mathematics (Izvestiya VUZ. Matematika), 1991, 35:8, 63–66 He also developed a very strong method with many page tables to sum series based on it (reference [4] in ...

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Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is $$f(x) = \ln(1-\cos(x)) = -\ln(2) + \sum_{n=1}^\infty \dfrac{2 \cos(nx)}{n}$$ where the series diverges at multiples of $2\pi$. EDIT: For Fejér's example of a ...

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