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Polynomial approximation of an entire function and rate of growth of Taylor coefficients

Published online by Cambridge University Press:  20 January 2009

M. Freund  and
E. Görlich
E. Görlich
Affiliation:
Lehrstuhl A für Mathematik, Technological University of Aachen, 5100 Aachen, Fed. Rep. Germany
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The best uniform approximation of a function f on [-1,1] by real algebraic polynomials satisfies

if and only if ƒ is the restriction to [- 1,1] of an entire function (Bernstein [2], p. 113, see also [12], pp. 83–85). For such functions ƒ the rate of best approximation has been characterized by Varga [24], Reddy [24], Shah [21], and Kapoor and Nautiyal [10] in terms of order and type of ƒ , lower order and type, and in terms of more general concepts of order. On the other hand, order and type of ƒ are connected with the Taylor coefficients, i.e. with the rate of growth of the sequence (see [23], p. 41 or [3], pp. 11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequence are obtained. But in such results the rate ofbest approximation is always described by a limit relation, e.g. of the form , and this causes a considerable loss of precision, as will be discussed in more detail in Section 3 (in this respect cf. also the remark by Bernstein [2], pp. 114/115).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 3 , October 1985 , pp. 341 - 348
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

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