Editing Distribution (mathematics) (section)

====Convolution versus multiplication====

In general, [[Regularization (physics)|regularity]] is required for multiplication products, and [[Principle of locality|locality]] is required for convolution products. It is expressed in the following extension of the [[Convolution theorem|Convolution Theorem]] which guarantees the existence of both convolution and multiplication products. Let <math>F(\alpha) = f \in \mathcal{O}'_C</math> be a rapidly decreasing tempered distribution or, equivalently, <math>F(f) = \alpha \in \mathcal{O}_M</math> be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let <math>F</math> be the normalized (unitary, ordinary frequency) [[Fourier transform]].<ref>{{cite book|last=Folland|first=G.B.|title=Harmonic Analysis in Phase Space|publisher=Princeton University Press|publication-place=Princeton, NJ|year=1989}}</ref> Then, according to {{harvtxt|Schwartz|1951}},
<math display=block>F(f * g) = F(f) \cdot F(g) \qquad \text{ and } \qquad F(\alpha \cdot g) = F(\alpha) * F(g)</math>
hold within the space of tempered distributions.<ref>{{cite book|last=Horváth|first=John|author-link = John Horvath (mathematician)|title=Topological Vector Spaces and Distributions|publisher=Addison-Wesley Publishing Company|publication-place=Reading, MA|year=1966}}</ref><ref>{{cite book|last=Barros-Neto|first=José|title=An Introduction to the Theory of Distributions|publisher=Dekker|publication-place=New York, NY|year=1973}}</ref><ref>{{cite book|last=Petersen|first=Bent E.|title=Introduction to the Fourier Transform and Pseudo-Differential Operators|publisher=Pitman Publishing|publication-place=Boston, MA|year=1983}}</ref> In particular, these equations become the [[Poisson summation formula|Poisson Summation Formula]] if <math>g \equiv \operatorname{\text{Ш}}</math> is the [[Dirac comb|Dirac Comb]].<ref>{{cite book|last=Woodward|first=P.M.|title=Probability and Information Theory with Applications to Radar|publisher=Pergamon Press|publication-place=Oxford, UK|year=1953}}</ref> The space of all rapidly decreasing tempered distributions is also called the space of {{em|convolution operators}} <math>\mathcal{O}'_C</math> and the space of all ordinary functions within the space of tempered distributions is also called the space of {{em|multiplication operators}} <math>\mathcal{O}_M.</math> More generally, <math>F(\mathcal{O}'_C) = \mathcal{O}_M</math> and <math>F(\mathcal{O}_M) = \mathcal{O}'_C.</math>{{sfn|Trèves|2006|pp=318-319}}<ref>{{cite book|last1=Friedlander|first1=F.G.|last2=Joshi|first2=M.S.|title=Introduction to the Theory of Distributions|publisher=Cambridge University Press|publication-place=Cambridge, UK|year=1998}}</ref> A particular case is the [[Paley–Wiener theorem#Schwartz's Paley–Wiener theorem|Paley-Wiener-Schwartz Theorem]] which states that <math>F(\mathcal{E}') = \operatorname{PW}</math> and <math>F(\operatorname{PW} ) = \mathcal{E}'.</math> This is because <math>\mathcal{E}' \subseteq \mathcal{O}'_C</math> and <math>\operatorname{PW} \subseteq \mathcal{O}_M.</math> In other words, compactly supported tempered distributions <math>\mathcal{E}'</math> belong to the space of {{em|convolution operators}} <math>\mathcal{O}'_C</math> and
Paley-Wiener functions <math>\operatorname{PW},</math> better known as [[Bandlimiting|bandlimited functions]], belong to the space of {{em|multiplication operators}} <math>\mathcal{O}_M.</math>{{sfn|Schwartz|1951}}

For example, let <math>g \equiv \operatorname{\text{Ш}} \in \mathcal{S}'</math> be the Dirac comb and <math>f \equiv \delta \in \mathcal{E}'</math> be the [[Dirac delta function|Dirac delta]];then <math>\alpha \equiv 1 \in \operatorname{PW}</math> is the function that is constantly one and both equations yield the [[Dirac comb#Dirac-comb identity|Dirac-comb identity]]. Another example is to let <math>g</math> be the Dirac comb and <math>f \equiv \operatorname{rect} \in \mathcal{E}'</math> be the [[rectangular function]]; then <math>\alpha \equiv \operatorname{sinc} \in \operatorname{PW}</math> is the [[sinc function]] and both equations yield the [[Nyquist–Shannon sampling theorem|Classical Sampling Theorem]] for suitable <math>\operatorname{rect}</math> functions. More generally, if <math>g</math> is the Dirac comb and <math>f \in \mathcal{S} \subseteq \mathcal{O}'_C \cap \mathcal{O}_M</math> is a [[Smoothness|smooth]] [[window function]] ([[Schwartz space|Schwartz function]]), for example, the [[Gaussian function|Gaussian]], then <math>\alpha \in \mathcal{S}</math> is another smooth window function (Schwartz function). They are known as [[mollifier]]s, especially in [[partial differential equation]]s theory, or as [[Regularization (mathematics)|regularizers]] in [[Regularization (physics)|physics]] because they allow turning [[generalized function]]s into [[Function (mathematics)|regular functions]].