TY - JOUR

T1 - Functions with strictly decreasing distances from increasing Tchebycheff subspaces

AU - Amir, D.

AU - Ziegler, Z.

PY - 1972/10

Y1 - 1972/10

N2 - Let {ui}i = 0∞ be a sequence of continuous functions on [0, 1] such that (u0,..., uk) is a Tchebycheff system on [0, 1] for all k ≥ 0 and let C(u0,..., uk) denote the corresponding generalized convexity cone. It is proved that if f belongs to C(u0,..., un - 1), then its distance from the linear space spanned by (u0,..., un) is strictly smaller than its distance from the linear space spanned by (u0,..., un - 1). Other properties of the best approximants to such functions are also given. It is shown, by a general category argument, that no direct converse can exit. It is then established that if strict decrease of distances (or one of a number of other properties of the best approximants) holds for all subintervals of [0, 1], then f ε{lunate} C(u0,..., un - 1) for all of these.

AB - Let {ui}i = 0∞ be a sequence of continuous functions on [0, 1] such that (u0,..., uk) is a Tchebycheff system on [0, 1] for all k ≥ 0 and let C(u0,..., uk) denote the corresponding generalized convexity cone. It is proved that if f belongs to C(u0,..., un - 1), then its distance from the linear space spanned by (u0,..., un) is strictly smaller than its distance from the linear space spanned by (u0,..., un - 1). Other properties of the best approximants to such functions are also given. It is shown, by a general category argument, that no direct converse can exit. It is then established that if strict decrease of distances (or one of a number of other properties of the best approximants) holds for all subintervals of [0, 1], then f ε{lunate} C(u0,..., un - 1) for all of these.

UR - http://www.scopus.com/inward/record.url?scp=49649143180&partnerID=8YFLogxK

U2 - 10.1016/0021-9045(72)90067-6

DO - 10.1016/0021-9045(72)90067-6

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AN - SCOPUS:49649143180

SN - 0021-9045

VL - 6

SP - 332

EP - 344

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 3

ER -