TY - JOUR

T1 - A theorem of gopengauz type with added interpolatory conditions

AU - Kilgore, T.

AU - Prestin, J.

PY - 1994/1/1

Y1 - 1994/1/1

N2 - The theorem of Gopengauz guarantees the existence of a polynomial which well approximates a function f εCq [— 1,1], while at the same time its kth derivative (k ≤ q) well approximates the kth derivative of the function, and moreover the polynomial and its derivatives respectively interpolate the function and its derivatives at ±1. With more generality, we shall prescribe that the polynomial interpolate the function at up to q+1 points near 1 and up to q + 1 points near —1. The points may coalesce, in which case one also interpolates at the coalescent point a number of derivatives one less than the multiplicity of coalescence. Aside from intrinsic theoretical interest, our results are clearly applicable in describing more precisely the error incurred in certain linear processes of simultaneous approximation, such as interpolation with added nodes near ±1. The original theorem of Gopengauz will be shown to follow as a special case.

AB - The theorem of Gopengauz guarantees the existence of a polynomial which well approximates a function f εCq [— 1,1], while at the same time its kth derivative (k ≤ q) well approximates the kth derivative of the function, and moreover the polynomial and its derivatives respectively interpolate the function and its derivatives at ±1. With more generality, we shall prescribe that the polynomial interpolate the function at up to q+1 points near 1 and up to q + 1 points near —1. The points may coalesce, in which case one also interpolates at the coalescent point a number of derivatives one less than the multiplicity of coalescence. Aside from intrinsic theoretical interest, our results are clearly applicable in describing more precisely the error incurred in certain linear processes of simultaneous approximation, such as interpolation with added nodes near ±1. The original theorem of Gopengauz will be shown to follow as a special case.

UR - http://www.scopus.com/inward/record.url?scp=84972939428&partnerID=8YFLogxK

U2 - 10.1080/01630569408816596

DO - 10.1080/01630569408816596

M3 - Journal articles

AN - SCOPUS:84972939428

SN - 0163-0563

VL - 15

SP - 859

EP - 868

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

IS - 7-8

ER -