## New answers tagged fourier-analysis

0

Let $e_n:=\exp(\pi i n t)$ and by $[(e_n)_{n\in S}]$ denote the closed linear span.
Claim. $[(e_n)_{n\in S}]=L_2[-1/2, 1/2]$ if and only if $e_{\lambda}\in [(e_n)_{n\in S}]$ for some $\lambda \in S^c$.
Indeed, suppose $e_{\lambda}\in [(e_n)_{n\in S}]$ for some $\lambda \in S^c$. Then dividing by $e_{\lambda}$ we see that 1 can be approximated by linear ...

5

A more complete list of particular self-reciprocal Fourier functions, i.e. eigenfunctions of the cosine Fourier transform:
$1.$ $\displaystyle \frac{1}{\sqrt{x}}$
$2.$ $\displaystyle e^{-x^2/2}$ (and more generally $e^{-x^2/2}H_{2n}(x)$, where $H_n$ is Hermite polynomial)
$3.$ $\displaystyle\frac{1}{\cosh\sqrt{\frac{\pi}{2}}x}$
$4.$ $\displaystyle ...

0

This turns out to be too long for a comment.
I am not sure if one can say much in the discrete two-dimensional case, but something is known in one-dimension for continuous signals:
There is a theorem of Logan that a signal with Fourier-transform supported in one octave (i.e. in some $[-2B,-B]\cup[B,2B]$) is characterized (up to a multiplicative constant) ...

0

This is closely related to the so called Muntz-Szasz phenomenon. Suppose we are given an increasing sequence of positive real numbers
$$ (R):\;\; 0 <r_0<r_1<\cdots $$
We denote by $C_R([0,1])$ the vector subspace of $C([0,1])$ spanned by the "monomials'' $x^{r_k}$, $k\geq 0$. Then the Muntz-Szasz theorem says that $C_R([0,1])$ is dense in ...

3

This is closely related to the so-called metric of strict convergence which is
$$
d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)|
$$
where $TV(u)$ denoted the total variation of $u$. This is indeed a metric on the space $BV(\Omega)$ (also for $\Omega\subset\mathbb{R}^n$). Hence, strict convergence of $u_n$ to $u$ is nothing else than saying
$$
u_n\to ...

3

Fourier dimension doesn't directly say anything about lower bounds for the mass of balls. Fourier dimension is smaller or equal than Hausdorff dimension, and in order to give a bound of the form $\mu(B(x,r))\ge r^s$, the exponent $s$ needs to be large.
This suggests that one should seek a notion of dimension that gives larger values than Hausdorff ...

4

Regarding Q1, if one selects $v$ randomly and uses Khintchines's inequality one obtains a lower bound $f(n) \gg n$, which complements the trivial upper bound of $f(n) \leq n$ coming from Plancherel and Cauchy-Schwarz. It looks like the same argument should also show that $f(n,w)$ is comparable to $w^{1/2} n^{1/2}$ for Q2.
For Q3, Theorem 1.3 of this paper ...

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