English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The Howe dual pair (sl(2),O(m)) allows the characterization of the classicalFourier transform (FT) on the space of rapidly decreasing functions as theexponential of a well-chosen element of sl(2) such that the Helmholtz relationsare satisfied. In this paper we first investigate what happens when instead weconsider exponentials of elements of the universal enveloping algebra of sl(2).This leads to a complete class of generalized Fourier transforms, that allsatisfy properties similar to the classical FT. There is moreover a finitesubset of transforms which very closely resemble the FT. We obtain operatorexponential expressions for all these transforms by making extensive use of thetheory of integer-valued polynomials. We also find a plane wave decompositionof their integral kernel and establish uncertainty principles. In importantspecial cases we even obtain closed formulas for the integral kernels. In thesecond part of the paper, the same problem is considered for the dual pair(osp(1|2),Spin(m)), in the context of the Dirac operator. This connects ourresults with the Clifford-Fourier transform studied in previous work.

Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)