Nonperiodic sampling and the local three squares theorem

Nonperiodic sampling and reconstruction from averages

In this paper, we show that recovery of a function from its averages over squares in the plane is closely related to a problem of recovery of bandlimited functions from samples on unions of regular lattices. We use this observation to construct explicit solutions to the Bezout equation which can be easily implemented in software. We also show that these sampling results give a new proof of the ...

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

The Sampling Theorem and the Bandpass Theorem

Nonperiodic Sampling of Bandlimited Functions on Unions of Rectangular Lattices

It is shown that a function f 2 L p ?R; R], 1 p < 1, is completely determined by the samples of ^ f on sets = m i=1 fn=2r i g n2Z where R = P r i , and r i =r j is irrational if i 6 = j, and of ^ f (j) (0) for j = 1; : : :; m ? 1. If f 2 C m?2?k ?R; R], then the samples of ^ f on and only the rst k derivatives of ^ f at 0 are required to completely determine f. Higher dimensional analogues of t...

A Theorem of Minkowski; the Four Squares Theorem

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted...