Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T22:55:38.634Z Has data issue: false hasContentIssue false
  • English
  • Français

An Analogue of a Problem of J. Balázs and P. Turán

Published online by Cambridge University Press:  20 November 2018

J. Prasad  and
A. K. Varma
J. Prasad
Affiliation:
University of Florida, Gainesville, Florida
A. K. Varma
Affiliation:
University of Florida, Gainesville, Florida
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.

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.

Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 54 - 63
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Balâzs, J. and Turân, P., Notes on interpolation. II, Acta Math. Acad. Sci. Hungar. 8 (1957), 201–215 Google Scholar
2. Balâzs, J. and Turân, P., Notes on interpolation. III, Acta Math. Acad. Sci. Hungar. 9 (1958), 195–214 Google Scholar
3. Balâzs, J. and Turân, P., Notes on interpolation. IV, Acta Math. Acad. Sci. Hungar. 9 (1958), 243–258 Google Scholar
4. Egervâry, E. and Turân, P., Notes on interpolation. V, Acta Math. Acad. Sci. Hungar. 9 (1958), 259–267 Google Scholar
5. Freud, G., Bemerkung iiber die Konvergenz eines interpolations-verjdhrens von P. Turân, 9 (1958), 337–341.Google Scholar
6. Sansone, G., Orthogonal functions, Vol. IX (Interscience, New York, 1959).Google Scholar
7. Saxena, R. B., On some interpolatory properties of Legendre polynomials. III, Bull. Inst. Math. 8 (1964), 63–93 Google Scholar
8. Surânyi, J. and Turân, P., Notes on interpolation. I, Acta Math. Acad. Sci. Hungar. 6 (1955), 67–79 Google Scholar
9. Szegô, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23 (Amer. Math. Soc, Providence, R.I., 1959).Google Scholar
10. Varma, A. K., On a problem of P. Turân on lacunary interpolation, Can. Math. Bull. 10 (1967), 531–557.Google Scholar
11. Lacunary interpolation through trigonometric and power polynomials (Ph.D. Thesis, University of Alberta, Calgary, Alberta, 1964).Google Scholar