Titchmarsh convolution theorem




The Titchmarsh convolution theorem is named after Edward Charles Titchmarsh,
a British mathematician. The theorem describes the properties of the support of the convolution of two functions.



Titchmarsh convolution theorem


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



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


∫0xφ(t)ψ(x−t)dt=0{displaystyle int _{0}^{x}varphi (t)psi (x-t),dt=0}{displaystyle int _{0}^{x}varphi (t)psi (x-t),dt=0}

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



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



Let φ,ψ∈L1(R){displaystyle varphi ,psi in L^{1}(mathbb {R} )}{displaystyle varphi ,psi in L^{1}(mathbb {R} )}. Then infsupp⁡φ∗ψ=infsupp⁡φ+infsupp⁡ψ{displaystyle inf operatorname {supp} varphi ast psi =inf operatorname {supp} varphi +inf operatorname {supp} psi }{displaystyle inf operatorname {supp} varphi ast psi =inf operatorname {supp} varphi +inf operatorname {supp} psi } if the right-hand side is finite.

Similarly, supsupp⁡φ∗ψ=supsupp⁡φ+supsupp⁡ψ{displaystyle sup operatorname {supp} varphi ast psi =sup operatorname {supp} varphi +sup operatorname {supp} psi }{displaystyle sup operatorname {supp} varphi ast psi =sup operatorname {supp} varphi +sup operatorname {supp} psi } if the right-hand side is finite.


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 }{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 proved by
J.-L. Lions in 1951:


If φ,ψ∈E′(Rn){displaystyle varphi ,psi in {mathcal {E}}'(mathbb {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 .}{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.} }{displaystyle operatorname {c.h.} } denotes the convex hull of the set. E′(Rn){displaystyle {mathcal {E}}'(mathbb {R} ^{n})}{displaystyle {mathcal {E}}'(mathbb {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).



References



  • Titchmarsh, E.C. (1926). "The zeros of certain integral functions". Proceedings of the London Mathematical Society. 25: 283–302. doi:10.1112/plms/s2-25.1.283..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Lions, J.-L. (1951). "Supports de produits de composition". Les Comptes rendus de l'Académie des sciences (I and II)|format= requires |url= (help). 232: 1530–1532, 1622–1624.


  • Mikusiński, J. and Świerczkowski, S. (1960). "Titchmarsh's theorem on convolution and the theory of Dufresnoy". Prace Matematyczne. 4: 59–76.CS1 maint: Uses authors parameter (link)


  • Yosida, K. (1980). Functional Analysis. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.). Berlin: Springer-Verlag.


  • Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators, I. Springer Study Edition (2nd ed.). Berlin: Springer-Verlag.


  • Levin, B. Ya. (1996). Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. Providence, RI: American Mathematical Society.



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