Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-08T05:33:44.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Variation-Diminishing Transformations and General Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

I. I. Hirschman JR.
I. I. Hirschman JR.*
Affiliation:
Washington University St. Louis, Missouri
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let α(dx) be a finite measure defined on the Borel subsets of [—1, 1], the spectrum of which is infinite. Let be the family of orthonormal polynomials associated with a, so that

The are uniquely determined by this and by the condition

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 98 - 107
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Edrei, A., On the generating functions of a doubly infinite totally positive sequence, Trans. Amer. Math. Soc, 74 (1953), 367–383.Google Scholar
2. Gantmacher, F. R. and Krein, M. G., Oszillationsmatrizen, Oszillationskerne unci kleine Schwingungen mechanischer Système (Berlin, 1960).Google Scholar
3. Grenander, U. and Szegô, G., Toeplitz forms and their applications (Berkeley and Los Angeles, 1958).Google Scholar
4. Hirschman, I. I. Jr., Variation diminishing Hankel transforms, J. Analyse Math., 8 (1960- 61), 307–336.Google Scholar
5. Hirschman, I. I. Jr., Variation diminishing transformations and ultraspherical polynomials, J. Analyse Math., 8 (1960-61), 337–360.Google Scholar
6. Hirschman, I. I. Jr., Variation diminishing transformations and orthogonal polynomials, J. Analyse Math., 9 (1961), 177–193.Google Scholar
7. Hirschman, I. I. Jr., Variation diminishing transformations and Sturm-Liouville systems, Comment. Math. Helv., 86 (1961), 214–233.Google Scholar
8. Hirschman, I. I. Jr., and Widder, D. V., The convolution transform (Princeton, 1955).Google Scholar
9. Karlin, S. and McGregor, J., The differential equations of birth-and-death processes and the Stieltjes moment problem, Trans. Amer. Math. Soc, 85 (1957), 489–546.Google Scholar
10. Schoenberg, I. J., On Polya frequency functions, J. Analyse Math., 1 (1951), 331–374.Google Scholar
11. Shohat, J. A. and Tamarkin, J. D., The problem of moments (New York, 1943).Google Scholar
12. Szegô, G., Orthogonal polynomials (New York, 1939).Google Scholar