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7

Put $f(z) = \sum_{n=0}^{\infty} z^{2^n}$. First note that the set of $\theta \in [0,1)$ such that the binary expansion of $\theta$ has arbitrarily large strings of $100$ consecutive zeros is a set of measure $1$. Now take such a $\theta$, and let $N$ be such that $\{2^{N} \theta\} \le 2^{-100}$ (here $\{x\}$ stands for the fractional part of $x$). ...

6

I don't know any simply way, but I would be interested in one, too. In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes ...

0

See Katznelson's Harmonic analysis (page 61) for a very simple example of a continuous function whose fourier series diverges at a dense set of points. I believe du Bois Reymond is the person responsible for the first such example.

2

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 ...

3

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 ...

0

This is only a very partial answer. Suppose that $a<-n$ and $p$ is a non-integer in the interval $(-1,n)$. Then $2j-n-a>0$ for all $j=0,\dots,n$. So, by the mean-value theorem applied (say, repeatedly) to the $n$-fold symmetric difference in the expression of $I_{n,p,a}$ in $(**)$ in the question statement, one has I_{n,p,a}=i^p\, \Gamma (-p)\, ...

3

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.

5

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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