هم‌گشت: تفاوت میان نسخه‌ها

A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that ''S'' must be a [[continuous linear operator]] with respect to the appropriate [[مکان‌شناسی]]. It is known, for instance, that every continuous translation invariant continuous linear operator on ''L''¹ is the convolution with a finite [[Borel measure]]. More generally, every continuous translation invariant continuous linear operator on ''L''^''p'' for 1 ≤ ''p'' < ∞ is the convolution with a [[توزیع (ریاضیات)]] whose [[تبدیل فوریه]] is bounded. To wit, they are all given by bounded [[Fourier multiplier|Fourier multipliers]].

 

If ''G'' is a suitable [[گروه (ریاضی)|group]] endowed with a [[نظریه اندازه|measure]] λ, and if ''f'' and ''g'' are real or complex valued [[Lebesgue integral|integrable]] functions on ''G'', then we can define their convolution by