TY - JOUR

T1 - Convergence properties of sequences of functions with application to restricted derivative approximation

AU - Kimchi, E.

AU - Richter-Dyn, N.

PY - 1978/4

Y1 - 1978/4

N2 - Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a "monotone norm" (e.g., Lp) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k - 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.

AB - Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a "monotone norm" (e.g., Lp) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k - 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.

UR - http://www.scopus.com/inward/record.url?scp=49349126597&partnerID=8YFLogxK

U2 - 10.1016/0021-9045(78)90040-0

DO - 10.1016/0021-9045(78)90040-0

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AN - SCOPUS:49349126597

SN - 0021-9045

VL - 22

SP - 289

EP - 303

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 4

ER -