Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem

A Theorem concerning Systems of Residue Classes

By system we mean a multi-set whose elements are unordered but may occur repeatedly. FollowingŠ. Znám [8] we use a(n) to denote the residue class { x ∈ Z : x ≡ a (mod n) }. For a system (1) A = {a i (n i)} k i=1 of residue classes, the n i are called its moduli. Definition. An integer T is said to be a covering period of (1) if it is a period of the characteristic function of the set k i=1 a i ...

The Sampling Theorem and the Bandpass Theorem

The Norm Residue Isomorphism Theorem

We provide a patch to complete the proof of the Voevodsky-Rost Theorem, that the norm residue map is an isomorphism. (This settles the motivic Bloch-Kato conjecture).

Boas' Formula and Sampling Theorem

In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon sampling theorem; (ii) to present a generalized Parseval formula, which is suggested by the Boas’ formula; and (iii) to show that this provides a very smart derivation o...

The residue theorem and its applications

This text contains some notes to a three hour lecture in complex analysis given at Caltech. The lectures start from scratch and contain an essentially self-contained proof of the Jordan normal form theorem, I had learned from Eugene Trubowitz as an undergraduate at ETH Zürich in the third semester of the standard calculus education at that school. My text also includes two proofs of the fundame...