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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral analysis for the generalized Hermite polynomials


Author: Allan M. Krall
Journal: Trans. Amer. Math. Soc. 344 (1994), 155-172
MSC: Primary 33C45; Secondary 34B30, 34L10, 47B50, 47E05
MathSciNet review: 1242783
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Abstract: The operator theory associated with the Hermite polynomials does not extend to the generalized Hermite polynomials because the even and odd polynomials satisfy different differential equations. We show that this leads to two problems, each of interest on its own. We then weld them together to form a united spectral expansion. In addition, the exponent $ \mu $ in the weight $ \vert x{\vert^{2\mu }}{e^{ - {x^2}}}$ has traditionally always been greater than $ - \frac{1}{2}$. We show what happens if $ \mu \leq - \frac{1}{2}$. Finally, we examine the differential equations in left-definite spaces.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1242783-9
PII: S 0002-9947(1994)1242783-9
Article copyright: © Copyright 1994 American Mathematical Society