On some open problems of P. Turán concerning Birkhoff interpolation

Author:
A. K. Varma

Journal:
Trans. Amer. Math. Soc. **274** (1982), 797-808

MSC:
Primary 41A05

MathSciNet review:
675080

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Abstract: In 1974 P. Turán (see [**6**]) raised many interesting open problems in Approximation Theory some of which are on Birkhoff Interpolation. The object of this paper is to answer some of these problems (XXXVI-XXXIX). We obtain some new quadrature formulas where function values and second derivatives are only prescribed on the zeros of

**[1]**J. Balázs and P. Turán,*Notes on interpolation. II. Explicit formulae*, Acta Math. Acad. Sci. Hungar.**8**(1957), 201–215. MR**0088590****[2]**J. Balázs and P. Turán,*Notes on interpolation. III. Convergence*, Acta Math. Acad. Sci. Hungar.**9**(1958), 195–214. MR**0098264****[3]**George David Birkhoff,*General mean value and remainder theorems with applications to mechanical differentiation and quadrature*, Trans. Amer. Math. Soc.**7**(1906), no. 1, 107–136. MR**1500736**, 10.1090/S0002-9947-1906-1500736-1**[4]**G. G. Lorentz and S. D. Riemenschneider,*Birkhoff interpolation*.**[5]**J. Surányi and P. Turán,*Notes on interpolation. I. On some interpolatorical properties of the ultraspherical polynomials*, Acta Math. Acad. Sci. Hungar.**6**(1955), 67–80 (English, with Russian summary). MR**0071569****[6]**P. Turán,*On some open problems of approximation theory*, J. Approx. Theory**29**(1980), no. 1, 23–85. P. Turán memorial volume; Translated from the Hungarian by P. Szüsz. MR**595512**, 10.1016/0021-9045(80)90138-0**[7]**P. O. H. Vértesi,*A problem of P. Turán (on the mean convergence of lacunary interpolation)*, Acta Math. Acad. Sci. Hungar.**26**(1975), 153–162. MR**0364952**

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0675080-2

Article copyright:
© Copyright 1982
American Mathematical Society