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Titchmarsh convolution theorem

If φ ( t ) {\textstyle \varphi (t)\,} and ψ ( t ) {\textstyle \psi (t)} are integrable functions, such that

φ ∗ ψ = ∫ 0 x φ ( t ) ψ ( x − t ) d t = 0 {\displaystyle \varphi *\psi =\int _{0}^{x}\varphi (t)\psi (x-t)\,dt=0}

almost everywhere in the interval 0 < x < κ {\displaystyle 0<x<\kappa \,} , then there exist λ ≥ 0 {\displaystyle \lambda \geq 0} and μ ≥ 0 {\displaystyle \mu \geq 0} satisfying λ + μ ≥ κ {\displaystyle \lambda +\mu \geq \kappa } such that φ ( t ) = 0 {\displaystyle \varphi (t)=0\,} almost everywhere in 0 < t < λ {\displaystyle 0<t<\lambda } and ψ ( t ) = 0 {\displaystyle \psi (t)=0\,} almost everywhere in 0 < t < μ . {\displaystyle 0<t<\mu .}

As a corollary, if the integral above is 0 for all x > 0 , {\textstyle x>0,} then either φ {\textstyle \varphi \,} or ψ {\textstyle \psi } is almost everywhere 0 in the interval [ 0 , + ∞ ) . {\textstyle [0,+\infty ).} Thus the convolution of two functions on [ 0 , + ∞ ) {\textstyle [0,+\infty )} cannot be identically zero unless at least one of the two functions is identically zero.

As another corollary, if φ ∗ ψ ( x ) = 0 {\displaystyle \varphi *\psi (x)=0} for all x ∈ [ 0 , κ ] {\displaystyle x\in [0,\kappa ]} and one of the function φ {\displaystyle \varphi } or ψ {\displaystyle \psi } is almost everywhere not null in this interval, then the other function must be null almost everywhere in [ 0 , κ ] {\displaystyle [0,\kappa ]} .

The theorem can be restated in the following form:

Let φ , ψ ∈ L 1 ( R ) {\displaystyle \varphi ,\psi \in L^{1}(\mathbb {R} )} . Then inf supp ⁡ φ ∗ ψ = inf supp ⁡ φ + inf supp ⁡ ψ {\displaystyle \inf \operatorname {supp} \varphi \ast \psi =\inf \operatorname {supp} \varphi +\inf \operatorname {supp} \psi } if the left-hand side is finite. Similarly, sup supp ⁡ φ ∗ ψ = sup supp ⁡ φ + sup supp ⁡ ψ {\displaystyle \sup \operatorname {supp} \varphi \ast \psi =\sup \operatorname {supp} \varphi +\sup \operatorname {supp} \psi } if the right-hand side is finite.

Above, supp {\displaystyle \operatorname {supp} } denotes the support of a function f (i.e., the closure of the complement of f−1(0)) and inf {\displaystyle \inf } and sup {\displaystyle \sup } denote the infimum and supremum. This theorem essentially states that the well-known inclusion supp ⁡ φ ∗ ψ ⊂ supp ⁡ φ + supp ⁡ ψ {\displaystyle \operatorname {supp} \varphi \ast \psi \subset \operatorname {supp} \varphi +\operatorname {supp} \psi } is sharp at the boundary.

The higher-dimensional generalization in terms of the convex hull of the supports was proven by Jacques-Louis Lions in 1951:²

If φ , ψ ∈ E ′ ( R n ) {\displaystyle \varphi ,\psi \in {\mathcal {E}}'(\mathbb {R} ^{n})} , then c . h . ⁡ supp ⁡ φ ∗ ψ = c . h . ⁡ supp ⁡ φ + c . h . ⁡ supp ⁡ ψ {\displaystyle \operatorname {c.h.} \operatorname {supp} \varphi \ast \psi =\operatorname {c.h.} \operatorname {supp} \varphi +\operatorname {c.h.} \operatorname {supp} \psi }

Above, c . h . {\displaystyle \operatorname {c.h.} } denotes the convex hull of the set and E ′ ( R n ) {\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})} denotes the space of distributions with compact support.

The original proof by Titchmarsh uses complex-variable techniques, and is based on the Phragmén–Lindelöf principle, Jensen's inequality, Carleman's theorem, and Valiron's theorem. The theorem has since been proven several more times, typically using either real-variable³⁴⁵ or complex-variable⁶⁷⁸ methods. Gian-Carlo Rota has stated that no proof yet addresses the theorem's underlying combinatorial structure, which he believes is necessary for complete understanding.⁹

References

1. Titchmarsh, E. C. (1926). "The Zeros of Certain Integral Functions". Proceedings of the London Mathematical Society. s2-25 (1): 283–302. doi:10.1112/plms/s2-25.1.283. http://doi.wiley.com/10.1112/plms/s2-25.1.283 ↩

2. Lions, Jacques-Louis (1951). "Supports de produits de composition". Comptes rendus. 232 (17): 1530–1532. /wiki/Comptes_rendus ↩

3. Doss, Raouf (1988). "An elementary proof of Titchmarsh's convolution theorem" (PDF). Proceedings of the American Mathematical Society. 104 (1). https://www.ams.org/journals/proc/1988-104-01/S0002-9939-1988-0958063-5/S0002-9939-1988-0958063-5.pdf ↩

4. Kalisch, G. K. (1962-10-01). "A functional analysis proof of titchmarsh's theorem on convolution". Journal of Mathematical Analysis and Applications. 5 (2): 176–183. doi:10.1016/S0022-247X(62)80002-X. ISSN 0022-247X. https://doi.org/10.1016%2FS0022-247X%2862%2980002-X ↩

5. Mikusiński, J. (1953). "A new proof of Titchmarsh's theorem on convolution". Studia Mathematica. 13 (1): 56–58. doi:10.4064/sm-13-1-56-58. ISSN 0039-3223. https://doi.org/10.4064%2Fsm-13-1-56-58 ↩

6. Crum, M. M. (1941). "On the resultant of two functions". The Quarterly Journal of Mathematics. os-12 (1): 108–111. doi:10.1093/qmath/os-12.1.108. ISSN 0033-5606. https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-12.1.108 ↩

7. Dufresnoy, Jacques (1947). "Sur le produit de composition de deux fonctions". Comptes rendus. 225: 857–859. /wiki/Comptes_rendus ↩

8. Boas, Ralph P. (1954). Entire functions. New York: Academic Press. ISBN 0-12-108150-8. OCLC 847696. {{cite book}}: ISBN / Date incompatibility (help) 0-12-108150-8 ↩

9. Rota, Gian-Carlo (1998-06-01). "Ten Mathematics Problems I will never solve". Mitteilungen der Deutschen Mathematiker-Vereinigung (in German). 6 (2): 45–52. doi:10.1515/dmvm-1998-0215. ISSN 0942-5977. S2CID 120569917. https://doi.org/10.1515%2Fdmvm-1998-0215 ↩