Editing Convolution (section)

== Domain of definition ==

The convolution of two complex-valued functions on {{math|'''R'''<sup>''d''</sup>}} is itself a complex-valued function on {{math|'''R'''<sup>''d''</sup>}}, defined by:

:<math>(f * g )(x) = \int_{\mathbf{R}^d} f(y)g(x-y)\,dy = \int_{\mathbf{R}^d} f(x-y)g(y)\,dy,</math>

and is well-defined only if {{mvar|f}} and {{mvar|g}} decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in {{mvar|g}} at infinity can be easily offset by sufficiently rapid decay in {{mvar|f}}. The question of existence thus may involve different conditions on {{mvar|f}} and {{mvar|g}}:

=== Compactly supported functions ===
If {{mvar|f}} and {{mvar|g}} are [[compact support|compactly supported]] [[continuous function]]s, then their convolution exists, and is also compactly supported and continuous {{harv|Hörmander|1983|loc=Chapter 1}}. More generally, if either function (say {{mvar|f}}) is compactly supported and the other is [[locally integrable function|locally integrable]], then the convolution {{math|''f''∗''g''}} is well-defined and continuous.

Convolution of {{mvar|f}} and {{mvar|g}} is also well defined when both functions are locally square integrable on {{math|'''R'''}} and supported on an interval of the form {{math|[''a'', +∞)}} (or both supported on {{math|[−∞, ''a'']}}).

=== Integrable functions ===
The convolution of {{mvar|f}} and {{mvar|g}} exists if {{mvar|f}} and {{mvar|g}} are both [[Lebesgue integral|Lebesgue integrable function]]s in [[Lp space|{{math|''L''<sup>1</sup>}}({{math|'''R'''<sup>''d''</sup>}})]], and in this case {{math|''f''∗''g''}} is also integrable {{harv|Stein|Weiss|1971|loc=Theorem 1.3}}. This is a consequence of [[Fubini's theorem#Tonelli's theorem|Tonelli's theorem]]. This is also true for functions in {{math|''L''<sup>1</sup>}}, under the discrete convolution, or more generally for the [[#Convolutions on groups|convolution on any group]].

Likewise, if {{math|''f'' ∈ ''L''<sup>1</sup>}}({{math|'''R'''<sup>''d''</sup>}})&nbsp; and &nbsp;{{math|''g'' ∈ ''L''<sup>''p''</sup>}}({{math|'''R'''<sup>''d''</sup>}})&nbsp; where {{math|1 ≤ ''p'' ≤ ∞}},&nbsp; then &nbsp;{{math|''f''*''g'' ∈ ''L''<sup>''p''</sup>}}({{math|'''R'''<sup>''d''</sup>}}),&nbsp; and

:<math>\|{f}* g\|_p\le \|f\|_1\|g\|_p.</math>

In the particular case {{math|''p'' {{=}} 1}}, this shows that {{math|''L''<sup>1</sup>}} is a [[Banach algebra]] under the convolution (and equality of the two sides holds if {{mvar|f}} and {{mvar|g}} are non-negative almost everywhere).
 
More generally, [[Young's convolution inequality|Young's inequality]] implies that the convolution is a continuous bilinear map between suitable {{math|''L''<sup>''p''</sup>}} spaces. Specifically, if {{math| 1 ≤ ''p'', ''q'', ''r'' ≤ ∞}} satisfy:

:<math>\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1,</math>

then

:<math>\left\Vert f*g\right\Vert_r\le\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in L^p,\ g\in L^q,</math>

so that the convolution is a continuous bilinear mapping from {{math|''L''<sup>''p''</sup>×''L''<sup>''q''</sup>}} to {{math|''L''<sup>''r''</sup>}}.
The Young inequality for convolution is also true in other contexts (circle group, convolution on {{math|'''Z'''}}). The preceding inequality is not sharp on the real line: when {{math| 1 < ''p'', ''q'', ''r'' < ∞}}, there exists a constant {{math|''B''<sub>''p'',''q''</sub> < 1}} such that:

:<math>\left\Vert f*g\right\Vert_r\le B_{p,q}\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in L^p,\ g\in L^q.</math>

The optimal value of {{math|''B''<sub>''p'',''q''</sub>}} was discovered in 1975<ref>{{cite journal
| first1=William | last1=Beckner | authorlink1=William Beckner (mathematician)
| year=1975
| title=Inequalities in Fourier analysis
| journal=Annals of Mathematics |series=Second Series
| volume=102
| issue=1 | pages=159–182
| doi=10.2307/1970980| jstor=1970980}}</ref> and independently in 1976,<ref>{{cite journal
| first1=Herm Jan | last1=Brascamp
| first2=Elliott H. | last2=Lieb | authorlink2=Elliott H. Lieb
| year=1976
| title=Best constants in Young's inequality, its converse, and its generalization to more than three functions
| journal=[[Advances in Mathematics]]
| volume=20
| issue=2
| pages=151–173
| doi=10.1016/0001-8708(76)90184-5 | doi-access=free}}</ref> see [[Brascamp–Lieb inequality]].

A stronger estimate is true provided {{math| 1 < ''p'', ''q'', ''r'' < ∞}}:
:<math>\|f * g\|_r\le C_{p,q}\|f\|_p\|g\|_{q,w}</math>
where <math>\|g\|_{q,w}</math> is the [[Lp space#Weak Lp|weak {{math|''L''<sup>''q''</sup>}}]] norm. Convolution also defines a bilinear continuous map <math>L^{p,w}\times L^{q,w}\to L^{r,w}</math> for <math>1< p,q,r<\infty</math>, owing to the weak Young inequality:<ref>{{harvnb|Reed|Simon|1975|loc=IX.4}}</ref>
:<math>\|f * g\|_{r,w}\le C_{p,q}\|f\|_{p,w}\|g\|_{r,w}.</math>

=== Functions of rapid decay ===
In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if ''f'' and ''g'' both decay rapidly, then ''f''∗''g'' also decays rapidly. In particular, if ''f'' and ''g'' are [[rapidly decreasing function]]s, then so is the convolution ''f''∗''g''. Combined with the fact that convolution commutes with differentiation (see [[#Properties]]), it follows that the class of [[Schwartz function]]s is closed under convolution {{harv|Stein|Weiss|1971|loc=Theorem 3.3}}.

=== Distributions ===
{{Main article|Distribution (mathematics)}}
If ''f'' is a smooth function that is [[Support (mathematics)#Compact support|compactly supported]] and ''g'' is a distribution, then ''f''∗''g'' is a smooth function defined by

:<math>\int_{\mathbb{R}^d} {f}(y)g(x-y)\,dy = (f*g)(x) \in C^\infty(\mathbb{R}^d) .</math>

More generally, it is possible to extend the definition of the convolution in a unique way with <math>\varphi</math> the same as ''f'' above, so that the associative law

:<math>f* (g* \varphi) = (f* g)* \varphi</math>

remains valid in the case where ''f'' is a distribution, and ''g'' a compactly supported distribution {{harv|Hörmander|1983|loc=§4.2}}.

=== Measures ===

The convolution of any two [[Borel measure]]s ''μ'' and ''ν'' of [[bounded variation]] is the measure <math>\mu*\nu</math> defined by {{harv|Rudin|1962}}
:<math>\int_{\mathbf{R}^d} f(x) \, d(\mu*\nu)(x) = \int_{\mathbf{R}^d}\int_{\mathbf{R}^d}f(x+y)\,d\mu(x)\,d\nu(y).</math>

In particular,
: <math>(\mu*\nu)(A) = \int_{\mathbf{R}^d\times\mathbf R^d}1_A(x+y)\, d(\mu\times\nu)(x,y),</math>

where <math>A\subset\mathbf R^d</math> is a measurable set and <math>1_A</math> is the [[indicator function]] of <math>A</math>.

This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L<sup>1</sup> functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.

The convolution of measures also satisfies the following version of Young's inequality
:<math>\|\mu* \nu\|\le \|\mu\|\|\nu\| </math>
where the norm is the [[total variation]] of a measure. Because the space of measures of bounded variation is a [[Banach space]], convolution of measures can be treated with standard methods of [[functional analysis]] that may not apply for the convolution of distributions.