Titchmarsh convolution theorem

The Titchmarsh Convolution Theorem is named after Edward Charles Titchmarsh, a British mathematician. It is well-known that the support of the convolution of two functions $f,\,g\in L^{(}\mathbb {R} )$ is inside the sum of their supports:

${\rm {supp}}\,f\ast g\subset \mathop {\rm {supp}} \,f+\mathop {\rm {supp}} \,g.$

The Titchmarsh Convolution Theorem states that this inclusion is sharp at the boundary; namely, assuming that f and g have compact supports,

$\sup \mathop {\rm {supp}} \,f\ast g=\sup \mathop {\rm {supp}} \,f+\sup \mathop {\rm {supp}} \,g$

and similarly for $\inf \mathop {\rm {supp}} \,f\ast g$.

The initial formulation of [1] was as follows:

Suppose $f,\,g$ are integrable on the interval $(0,T)$ and that the convolution $f\ast g(t)=0$ on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (0,T)} . Then there are nonnegative numbers $\alpha$, $\beta$ with $\alpha +\beta \geq T$ for which $f(x)=0$ for almost all $x\in (0,\alpha )$ and $g(x)=0$ for almost all $x\in (0,\beta )$.

Higher-dimensional generalization

The higher-dimensional generalization is in terms of the convex hull of the supports:

$\mathop {c.h.} \mathop {\rm {supp}} \,f\ast g=\mathop {c.h.} \mathop {\rm {supp}} \,f+\mathop {c.h.} \mathop {\rm {supp}} \,g$

where $\mathop {c.h.}$ denotes the convex hull of the set.

The statement remains valid for $f,\,g\in {\mathcal {E}}'(\mathbb {R} ^{n})$, which denotes the space of distributions with compact support.

Proofs

The original proof is contained in [1]. More proofs are contained in [2, Theorem 4.3.3] (Harmonic Analysis style), [3, Chapter VI] (Real Analysis style), and [4, Lecture 16, Theorem 5] (Complex Analysis style).

References

[1] E.C. Titchmarsh, The zeros of certain integral functions, Proc. of the London Math. Soc. 25 (1926), 283--302.

[2] K. Yosida, Functional analysis (sixth ed.), Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123, Springer-Verlag, Berlin, 1980.

[3] L. Hormander, The analysis of linear partial differential operators. I (second ed.). Springer Study Edition, Springer-Verlag, Berlin, 1990.

[4] B.Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996.