# The convolution of analytic functionals

The convolution of analytic functionals This paper is to define the convolution of analytic functionals in $$Z'$$ Z ′ by the Fourier transform and product in $${\mathcal D}'$$ D ′ . We show that a new exchange formula \begin{aligned}F(f) \, \overline{*} \, F(g) = 2 \pi F ( f \circ g ) \end{aligned} F ( f ) ∗ ¯ F ( g ) = 2 π F ( f ∘ g ) is satisfied. Several interesting examples of computing convolutions in $$Z'$$ Z ′ are presented based on the exchange formula, which have never been investigated before. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

# The convolution of analytic functionals

, Volume 27 (4) – Apr 26, 2018
14 pages

/lp/springer-journals/the-convolution-of-analytic-functionals-oyfl3z5RXC
Publisher
Springer Journals
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-018-0078-5
Publisher site
See Article on Publisher Site

### Abstract

This paper is to define the convolution of analytic functionals in $$Z'$$ Z ′ by the Fourier transform and product in $${\mathcal D}'$$ D ′ . We show that a new exchange formula \begin{aligned}F(f) \, \overline{*} \, F(g) = 2 \pi F ( f \circ g ) \end{aligned} F ( f ) ∗ ¯ F ( g ) = 2 π F ( f ∘ g ) is satisfied. Several interesting examples of computing convolutions in $$Z'$$ Z ′ are presented based on the exchange formula, which have never been investigated before.

### Journal

The Journal of AnalysisSpringer Journals

Published: Apr 26, 2018

### References

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