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L2-Theory of Riesz Transforms for Orthogonal Expansions

✍ Scribed by Adam Nowak; Krzysztof Stempak


Publisher
SP Birkhäuser Verlag Boston
Year
2006
Tongue
English
Weight
322 KB
Volume
12
Category
Article
ISSN
1069-5869

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We study different Sobolev spaces associated with multidimensional Laguerre expansions. To do this we establish an analogue of P.A. Meyer's multiplier theorem, prove some transference results between higher order Riesz-Hermite and Riesz-Laguerre transforms, and introduce fractional derivatives and i

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## Riesz fractional derivatives of a function, D α x f (x) (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, G

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Let f (z)=a 0 , 0 (z)+a 1 , 1 (z)+ } } } +a n , n (z) be a polynomial of degree n, given as an orthogonal expansion with real coefficients. We study the location of the zeros of f relative to an interval and in terms of some of the coefficients. Our main theorem generalizes or refines results due to