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Convergence of Interpolation to Transforms of Totally Positive Kernels

Published online by Cambridge University Press:  20 November 2018

N. Dyn  and
D. S. Lubinsky
N. Dyn
Affiliation:
Tel Aviv University, Tel Aviv, Israel
D. S. Lubinsky
Affiliation:
Centre for Advanced Computing and Decision Support, Pretoria, Republic of South Africa
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Convergence of exponential sums

that interpolate to Laplace transforms

(1.1)

have been studied by several authors [3, 6, 8, 15]. For rational functions that interpolate to Markov functions (also called Hamburger or Stieltjes Series or Hilbert Transforms)

(1.2)

far more detailed convergence results are available (see [10, 11, 16] and references therein). Both (1.1) and (1.2) are special cases of the transform

(1.3)

where K(x, t) is a strictly totally positive kernel.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 750 - 768
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Achieser, N. I., Theory of approximation (Ungar, New York, 1956).Google Scholar
2. Bojanov, B. D., Braess, D. and Dyn, N., Generalized Gaussian quadrature formulas, J. Approx. Th. 48 (1978), 335–353.Google Scholar
3. Borwein, P., Uniform approximation by polynomials with variable exponents, Can. J. Math. 55 (1983), 547–577.Google Scholar
4. Braess, D., Chebyshev approximation by γ-polynomials II, J. Approx. Th. 11 (1974), 16–37.Google Scholar
5. Braess, D. and Dyn, N., On the uniqueness of monosplines and perfect splines of least L1 and L2-norms, J. Analyse Math. 41 (1982), 217–233.Google Scholar
6. Braess, D. and Saff, E. B., On the degree of approximation of completely monotone functions by exponential sums, to appear.Google Scholar
7. Dyn, N., Generalized monosplines and optimal approximation, Constr. Approx. 1 (1985), 137–154.Google Scholar
8. Kammler, D. W., Prony's method for completely monotonie functions, J. Math. Anal. Appl. 57(1977), 560–570.Google Scholar
9. Karlin, S. and Studden, W. J., Tchebycheff systems: with applications in analysis and statistics (Wiley, New York, 1966).Google Scholar
10. Lopez, G., On the asymptotics of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants, Math. USSR. Sbornik 56 (1987), 207–219.Google Scholar
11. Magnus, Al., Toeplitz matrix techniques and convergence of complex weight Padé approximants, J. Comp. Appl. Math. 19 (1987), 23–38.Google Scholar
12. Markushevich, A. I., Theory of functions of a complex variable, Vol. 1 (Prentice-Hall, New Jersey, 1965).Google Scholar
13. Pinkus, A., A simple proof of the Hobby-Rice theorem, Proc. of the Amer. Math. Soc. 60 (1976), 82–84.Google Scholar
14. Pinkus, A., N-widths in approximation theory (Springer-Verlag, Berlin, 1980).Google Scholar
15. Sidi, A. and Lubinsky, D. S., Convergence of exponential interpolation for completely bounded functions, J. Approx. Th. 39 (1983), 185–201.Google Scholar
16. Stahl, H., Beitrâge zum Problem der Konvergenz von Padé Approximierenden, Ph.D. Thesis, Technical University, Berlin (1976).Google Scholar
17. Still, G., On the approximation of holomorphicfunctions by Muntz polynomials in an interval away from the origin, J. Approx. Th. 45 (1985), 167–193.Google Scholar
18. Walsh, J. L., Interpolation and approximation by rational functions in the complex domain, Vol. 20, 2nd edn. (American Mathematical Society Colloquium Publns., Providence, 1956).Google Scholar