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Questions tagged [lacunary-series]

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If $f$ is continuous and $\hat{f}$ is supported on $S$ then its Fourier series converges

In general $\scriptstyle\text{(even if it is hard to show a simple counterexample)}$ that $f$ is continuous doesn't mean its Fourier series converges everywhere. In this question it is shown that if $...
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1answer
75 views

A tricky limit involving exponential integrals

We define exponential integral according to https://en.wikipedia.org/wiki/Exponential_integral#Definition_by_Ein as $$\text{Ei}_n(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t^n} dt$$ I'm trying to ...
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A third way to solve linear functional equations inspired by lacunary series

Let $(f,\omega,H)$ be complex functions $ w \subseteq \mathbb{C} \rightarrow u \subseteq \mathbb{C}$ Then it's easy to see that a "formal" solution the following functional equation $$ f(\omega(x))...
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Is $\sum^\infty_{n=1}\mu(n)z^n$ a lacunary function?

Let $\mu(n)$ be the mobius function. Then, is $$f(z)=\sum^\infty_{n=1}\mu(n)z^n$$ a lacunary function? Clearly, the series converges in the open unit disk. Since I have read from somewhere (maybe ...
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Extending Lacunary Series beyond their disks

I've been for the past year and half fascinated by the lacunary series $f(z) = \sum_{n=0}^{\infty} z^{2^n}$. This function obeys the following equation inside the unit disk. $$f(z^2) = f(z)-z$$ And ...
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1answer
132 views

Is there a positive-semidefinite convolution kernel, that is continuous at $0$ but discontinuous elsewhere?

A positive-semidefinite, symmetric convolution kernel on the circle $\mathbb{T}^1$ is a function $k:\mathbb{T}^1\to\mathbb{R}$ such that $k(x)=k(-x)$, and $\sum_{i=1}^n\sum_{j=1}^n k(x_i-x_j)c_i c_j\...
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1answer
133 views

Convergence of a complex series on the boundary

I have the following complex power-series: $$\sum_{n=1}^\infty \frac{z^{n!}}{n} $$ Its radius of convergence is $R=1$. I am trying to investigate its behavior on the boundary ($z$ such that $|z|=1$). ...
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Lacunary functions and sums of reciprocals

Let $\Lambda=\left\{ \lambda_{n}\right\} _{n=0}^{\infty}$ be an infinite, strictly increasing sequence of non-negative integers. I say that $\Lambda$ is reciprocal-summable if: $$\sum_{\lambda\in\...
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1answer
81 views

How fast does $\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$ go to $\alpha$?

In this question: $\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$ it is established that $$\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right) = \...
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$\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$

Has anybody seen (or can anybody come up with) a proof that $$\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right) = \alpha$$ for all $\alpha > 0$? And also that $$\lim_{\...
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4answers
204 views

Lacunary series - Finding a limit

Let, $f$ be the function defined on the open unit disk: \begin{equation*} f(x)=\sum_{n=0}^{+ \infty} x^{n^2} \end{equation*} The aim of the exercise is to find the limit of $f(x)$ as $x$ approaches $-...
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Does $\sup_n r_2(E,n)<\infty$ imply $\Lambda(4)$?

A subset $E\subset \mathbb Z$ is call $\Lambda(4)$ set, if it has a const $C=C(E)<\infty$ st. for all sequence $a\in l^2(E)$, $\|\sum_{n\in E}a_ne^{inx}\|_{L^4}\le C\|a\|_{l^2}$. Denote $r_2(E,n)=\...
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2answers
148 views

If a periodic function with lacunary Fourier series is zero on a small interval then is it smooth?

Let $f\in L^{1}_{2\pi}=L(T)$ with $f\sim\sum a_{n}\cos(\lambda_{n}t)$ with $(\lambda_{n})$ lacunary and $f$ Hölder continuous on the circle $T$. Moreover, $a_{n}=O(\lambda^{-\alpha}_{n})$. Suppose ...
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1answer
321 views

Lacunary Fourier series and Hölder continuity at a point on the circle

Let $(\lambda_{n})$ be lacunary (i.e. $\exists$ constant $q>1$ such that $\lambda_{n+1}>q\lambda_{n}$ for all $n\in\mathbb{N}$); $f\in L^{1}(T)$ with Fourier series $\sum_{n\in\mathbb{N}}a_{n}\...