Titchmarsh convolution theorem

Titchmarsh convolution theorem

E.C. Titchmarsh proved the following theorem in 1926:

If ϕ ( t ) and ψ ( t ) are integrable functions, such that ∫ 0 x ϕ ( t ) ψ ( x − t ) d t = 0 almost everywhere in the interval 0 < x < κ , then there exist λ ≥ 0 and μ ≥ 0 satisfying λ + μ ≥ κ such that ϕ ( t ) = 0 almost everywhere in ( 0 , λ ) , and ψ ( t ) = 0 almost everywhere in ( 0 , μ ) .

This result, known as the Titchmarsh convolution theorem, can be restated in the following form:

Let ϕ , ψ ∈ L 1 ( R ) . Then inf supp ⁡ ϕ ∗ ψ = inf supp ⁡ ϕ + inf supp ⁡ ψ if the right-hand side is finite. Similarly, sup supp ⁡ ϕ ∗ ψ = sup supp ⁡ ϕ + sup supp ⁡ ψ if the right-hand side is finite.

This theorem essentially states that the well-known inclusion

supp ⁡ ϕ ∗ ψ ⊂ supp ⁡ ϕ + supp ⁡ ψ

is sharp at the boundary.

The higher-dimensional generalization in terms of the convex hull of the supports was proved by J.-L. Lions in 1951:

If ϕ , ψ ∈ E ′ ( R n ) , then c . h . ⁡ supp ⁡ ϕ ∗ ψ = c . h . ⁡ supp ⁡ ϕ + c . h . ⁡ supp ⁡ ψ .

Above, c . h . denotes the convex hull of the set E ′ ( R n ) denotes the space of distributions with compact support.

The theorem lacks an elementary proof. The original proof by Titchmarsh is based on the Phragmén–Lindelöf principle, Jensen's inequality, the Theorem of Carleman, and Theorem of Valiron. More proofs are contained in [Hörmander, Theorem 4.3.3] (harmonic analysis style), [Yosida, Chapter VI] (real analysis style), and [Levin, Lecture 16] (complex analysis style).