On the divergence of trigonometric lacunary interpolation

L. Imhof

doi:10.1007/bf00050687

On the divergence of trigonometric lacunary interpolation

Acta Mathematica Hungarica ◽

10.1007/bf00050687 ◽

1996 ◽

Vol 72 (3) ◽

pp. 243-250 ◽

Cited By ~ 1

Author(s):

L. Imhof

Keyword(s):

Lacunary Interpolation

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Some remarks on lacunary interpolation in the roots of unity

Israel Journal of Mathematics ◽

10.1007/bf02759733 ◽

1964 ◽

Vol 2 (1) ◽

pp. 41-49 ◽

Cited By ~ 9

Author(s):

A. Sharma

Keyword(s):

Roots Of Unity ◽

Lacunary Interpolation

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-006-2 ◽

1969 ◽

Vol 21 ◽

pp. 54-63 ◽

Cited By ~ 5

Author(s):

J. Prasad ◽

A. K. Varma

Keyword(s):

Differential Equation ◽

Modulus Of Continuity ◽

Legendre Polynomial ◽

Existence And Uniqueness ◽

Explicit Representation ◽

Lacunary Interpolation ◽

First Derivative ◽

Second Derivatives ◽

Significant Application

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.

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