Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T02:49:34.915Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  20 January 2009

E. Görlich
Affiliation:
Lehrstuhl A für Mathematik, Technological University of Aachen, 5100 Aachen, Fed. Rep. Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1.Bajpai, S. K., Gautam, S. K. S. and Bajpai, S. S., Generalization of growth constants, 1, Ann. Polon. Math. 27 (1980), 1324.CrossRefGoogle Scholar
2.Bernstein, S., Lecons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle (Gauthier-Villars, Paris, 1926).Google Scholar
3.Boas, R. P. Jr., Entire Functions (Academic Press, New York, 1954).Google Scholar
4.Freud, M. and Gorlich, E., On the relation between maximum modulus, maximum term, and Taylor coefficients of an entire function, J. Approx. Theory, 43 (1985), 194203.CrossRefGoogle Scholar
5.Golomb, M., Lectures on theory of approximation (Mimeographed Lecture Notes, Argonne National Laboratory, 1962).Google Scholar
6.Juneja, O. P., On the coefficients of an entire series of finite order, Arch. Math. 21 (1970), 374378.CrossRefGoogle Scholar
7.Juneja, O. P., On the coefficients of an entire series, J. Analyse Math. 24 (1971), 395401.CrossRefGoogle Scholar
8.Juneja, O. P. and Kapoor, G. P., On the lower order of entire functions J. London Math. Soc. (2) 5 (1972), 310312.CrossRefGoogle Scholar
9.Juneja, O. P., Kapoor, G. P. and Bajpai, S. K., On the (p.q)–type and lower (p.q)–type of an entire function, J. Reine Angew. Math. 290 (1977), 180189.Google Scholar
10.Kapoor, G. P. and Nautiyal, A., Polynomial approximation of an entire function of slow growth, J. Approx. Theory 32 (1981), 6475.CrossRefGoogle Scholar
11.Luke, Y. L., The special functions and their approximation, Vol. I. (Academic Press, New York, 1969).Google Scholar
12.Meinardus, G., Approximation von Funktionen und ihre numerische Behandlung (Springer, Berlin, 1964).CrossRefGoogle Scholar
13.Nandan, K., Doherey, R. P. and Srivastava, R. S. L., On the generalized type and generalized lower type of an entire function with index pair (p, q), Indian J. Pure Appl. Math. 11 (1980), 14241433.Google Scholar
14.Reddy, A. R., Approximation of an entire function, J. Approx. Theory 3 (1970), 128137.CrossRefGoogle Scholar
15.Reddy, A. R., Best polynomial approximation to certain entire functions, J. Approx. Theory 5 (1972), 97112.CrossRefGoogle Scholar
16.Sato, D., On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc. 69 (1963), 411414.CrossRefGoogle Scholar
17.Schonhage, A., Ober das Wachstum zusammengesetzter Funktionen, Math. Z. 73 (1960), 2244.CrossRefGoogle Scholar
18.Seremeta, M. N., On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion, Amer. Math. Soc. Transl. (2) 88 (1970), 291301.Google Scholar
19.Shah, S. M., On the lower order of integral functions, Bull. Amer. Math. Soc. 52 (1946), 10461052.CrossRefGoogle Scholar
20.Shah, S. M., On the coefficients of an entire series of finite order, J. London Math. Soc. 26 (1952), 4546.Google Scholar
21.Shah, S. M., Polynomial approximation of an entire function and generalized orders, J. Approx. Theory 19 (1977), 315324.CrossRefGoogle Scholar
22.Shah, S. M. and Ishaq, M., On the maximum modulus and the coefficients of an entire series, J. Indian Math. Soc. 16 (1952), 177182.Google Scholar
23.Valiron, G., Lectures on the general theory of integral functions (Chelsea Publ. Comp., New York, 1949).Google Scholar
24.Varga, R. S., On an extension of a result of S. N. Bernstein, J. Approx. Theory 1 (1968), 176179.CrossRefGoogle Scholar