The Fourier exchange theorem for kernels

Abstract

The Fourier exchange theorem

is generalized to 𝒮′-convolvable kernels, i.e. temperate distributions K(x, y) ∈ 𝒮′(ℝ²ⁿ ) for which K(x − y, y) is semitemperate with respect to x and partially summable with respect to y.

Keywords:

• temperate distributions
• convolution of distributions
• Fourier transform

AMS Subject Classifications::

• 46F10
• 46F12
• 42B10
• 42B20

Acknowledgement

I am indebted to my colleague Peter Wagner for fruitful discussions on the topics of this article.