Theorem

Titchmarsh Convolution Theorem

Analysis

The Titchmarsh Convolution Theorem describes the support of the convolution of two functions on the real line, showing that if the convolution vanishes on an interval near a given point, the two original functions must together leave a matching gap near their own combined starting points. Proved by Edward Charles Titchmarsh in 1926, it is a classical uniqueness result of analysis with consequences for integral equations and for questions of when a function can be represented as a Laplace transform.

Facts

Statement

Describes the support of the convolution of two functions: if integrable functions phi and psi have convolution vanishing almost everywhere on 0 < x < kappa, then there exist lambda >= 0 and mu >= 0 with lambda + mu >= kappa such that phi = 0 almost everywhere on (0, lambda) and psi = 0 almost everywhere on (0, mu). 1

Proof Year

1926 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Statement Form

Identity or Equation 1

Connections

Has Statement Form

In Branch

Harmonic Analysis

Source Titchmarsh convolution theorem (Wikipedia)

Proved By

Edward Charles Titchmarsh

Source Titchmarsh convolution theorem (Wikipedia)

Sources

1. Titchmarsh convolution theorem (Wikipedia)

Introduction, sentence 1
describes the properties of the support of the convolution of two functions
Introduction, sentence 2
It was proven by Edward Charles Titchmarsh in 1926
Proved By: Edward Charles Titchmarsh, Lead sentence, attribution
proven by Edward Charles Titchmarsh in 1926
In Branch: Harmonic Analysis, Lead section
theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in

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Titchmarsh Convolution Theorem

Has Statement Form

In Branch

Proved By