# Tagged Questions

0answers
147 views

3answers
437 views

### 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 ...
1answer
311 views

3answers
404 views

### 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 $f$ ...
0answers
173 views

### 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 ...
0answers
292 views

### 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 ...
0answers
99 views

### 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. ...
2answers
195 views

### 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 $\theta$ ...
1answer
345 views

### 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 ...
1answer
532 views

### 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, ...
2answers
712 views

2answers
1k views

### 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 ...
0answers
244 views

### 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 ...
4answers
4k views

### 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 ...
0answers
402 views

### 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 ...
2answers
540 views

### Convergence of squares of the moduli of partial sums of Fourier series

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 ...
1answer
631 views

### On the Existence of Certain Fourier Series

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 ?
2answers
2k views

### Positive-Definite Functions and Fourier Transforms

Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite. ...
1answer
425 views

### Is there any result about the uniform convergence rate of multi-dimensional Fourier series

For example in the 1-dimensional case, it is known that if f satisfies the α-Hölder condition, then $|f(x)-(S_Nf)(x)|\le K \frac{\ln N}{N^\alpha}$ where $S_N f$ is the n-term partial sum of the ...