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On Marcel Riesz's Ultrahyperbolic Kernel
Author(s) -
Trione Susana Elena
Publication year - 1988
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1988793185
Subject(s) - mathematics , dimension (graph theory) , kernel (algebra) , operator (biology) , constant (computer programming) , combinatorics , space (punctuation) , point (geometry) , mathematical analysis , pure mathematics , discrete mathematics , geometry , computer science , biochemistry , chemistry , repressor , transcription factor , gene , programming language , operating system
Let t = ( t 1 , …, t n ) be a point of ℝ n . We shall write . We put by definition R α ( u ) = u ( α − n )/2 / K n ( α ); here α is a complex parameter, n the dimension of the space, and K n ( α ) is a constant. First we evaluate □ R α ( u ) = R α ( u ), where □ the ultrahyperbolic operator. Then we obtain the following results: R −2 k ( u ) = □ k δ ; R 0 ( u ) = δ; and □ k R 2 k ( u ) = δ , k = 0, 1, …. The first result is the n ‐dimensional ultrahyperbolic correlative of the well‐known one‐dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.