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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 ...
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 ...