# Tagged Questions

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### A book about almost periodic functions [closed]

Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
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### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
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### In what ways is the standard Fourier basis optimal?

Is there any convincing sense in which the standard trigonometric basis for the space $V$ of square-integrable real-valued functions on $[-\pi,\pi]$ is optimal among all the orthonormal bases? (If ...
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### Simple closed curves and the coefficent of $\exp(i\theta)$ in the associated Fourier series

Given a continuous map $f:S^1\to \mathbb{C}$ from the unit circle to the complex numbers, one can form its Fourier series $\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those $f$ ...
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### Spectral gap of tempered distributions

Hi, Let $\Lambda\subset\mathbb{R}$ be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and $\delta_{\lambda}$ is a unit mass located at the point ...
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### Basic examples of induction on scales arguments

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth. Here is a ...
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### Differential equation with switched parameters and boundary conditions in integral form

Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem. ...
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### bounds on solution of an ODE

I am interested in getting a good bound for solution of the following ODE: $$f''(t) + n^2 f(t) = (\sin(\theta t))^n$$ with the boundary condition $f(0) = f'(0) = 0$ and $t \in [0,1]$, where $\theta$ ...
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### Stein's extension operator and wave front sets

Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein ...
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### An exponential polynomial with at least one bounded positivity component

In a forthcoming paper on nodal domains of Gaussian random functions, we (I and Misha Sodin) have a statement that is, roughly speaking, the following: if bounded nodal domains are possible at all, ...
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### Stone-Weierstrass theorem applied to Fourier series

This is a question on Fourier series convergence. The problem is, in the applications of the Stone Weierstrass approximation theorem on wikipedia, there's stated that as a consequence of the theorem ...
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### Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
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### Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
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### Phase perturbations in oscillatory integrals

I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in ...
Let $\mu$ be a complex measure on the unit circle. The Wiener theorem says that the sequence of the Cesaro means of $|\hat\mu_n|$ has a limit. Define $p_n(z)=\sum_{k=0}^n \hat\mu_k z^k$. Then the Abel ...
Is there an $f\in L^{1}(T)$ whose partial sums of Fourier series $S_{n}(f)$ satisfies $\|S_{n}(f)\|_{L^{1}(T)} \rightarrow \|f\|_{L^{1}(T)}$ but $S_{n}(f)$ fails to converge to $f$ in $L^1$-norm ?