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1 vote
1 answer
127 views

approximating differentiable functions with double trigonometric polynomials

Let $Q = [0,1]^2$. For sake of notation, let $$ f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi). $$ Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if $$ \|...
1 vote
0 answers
244 views

On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
4 votes
0 answers
104 views

Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials? In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
1 vote
0 answers
85 views

A kernel on the d-dimensional flat torus with smoothing properties in the $L^{\infty}$-norm

Let $\rho: \mathbb{R}^d\rightarrow \mathbb{R}_+$ be smooth, symmetric, of compact support, and satisfy $\int_{\mathbb{R}^d}\rho(x)dx=1$. For each $\epsilon>0$, set $\rho_{\epsilon}(x)=\epsilon^{-d}\...
2 votes
2 answers
161 views

Non-global oscillation of banded Fourier transform

Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function $f$ with support $[0, N]$ $$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$ Of ...
5 votes
2 answers
958 views

L1 distance from a trigonometric susbspace

How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in \mathbb{Z}\...