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4 votes
1 answer
191 views

Integral of $\ln(1/|f|)$ for $f$ bandlimited

I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...
pipenauss's user avatar
  • 319
1 vote
1 answer
132 views

Plancharel-Pólya inequality for functions of exponential type

If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$ then it can be proven that $$\int_{-\infty}^{+\...
pipenauss's user avatar
  • 319
1 vote
1 answer
167 views

Sampling set: relatively dense and uniformly discrete

The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$ We say that a discrete set $\Lambda\subset\...
pipenauss's user avatar
  • 319
3 votes
2 answers
236 views

Sampling Theorem for non-bandlimited Functions

The classical Shannon sampling theorem states that a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol $...
conan's user avatar
  • 41