| Title:
|
On the approximation of entire functions over Carathéodory domains (English) |
| Author:
|
Kumar, D. |
| Author:
|
Kasana, H. S. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
4 |
| Year:
|
1994 |
| Pages:
|
681-689 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $D$ be a Carathéodory domain. For $1\leq p\leq \infty $, let $L^p(D)$ be the class of all functions $f$ holomorphic in $D$ such that $\|f\|_{D,p}=[\frac{1}{A}\int\int_{D}^{}|f(z)|^p\,dx\,dy]^{1/p}<\infty $, where $A$ is the area of $D$. For $f\in L^p(D)$, set $$ E_n^p(f)=\inf _{t\in \pi _n} \|f-t\|_{D,p}\,; $$ $\pi _n$ consists of all polynomials of degree at most $n$. In this paper we study the growth of an entire function in terms of approximation error in $L^p$-norm on $D$. (English) |
| Keyword:
|
approximation error |
| Keyword:
|
generalized parameters |
| Keyword:
|
$L^p$ norm and Fourier coefficients |
| MSC:
|
30D15 |
| MSC:
|
30E10 |
| idZBL:
|
Zbl 0815.30019 |
| idMR:
|
MR1321238 |
| . |
| Date available:
|
2009-01-08T18:14:19Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118709 |
| . |
| Reference:
|
[1] Bernstein S.N.: Constructive theory of functions (1905-1930) in collected works, I.Akademia Nauk SSSR, 1952. MR 0048360 |
| Reference:
|
[2] Juneja O.P.: Approximation of an entire function.J. Approx. Theory 11 (1974), 343-349. Zbl 0285.30030, MR 0390608 |
| Reference:
|
[3] Kapoor G.P., Nautiyal A.: Polynomial approximation of an entire function of slow growth.J. Approx. Theory 32 (1981), 64-75. Zbl 0495.41005, MR 0629582 |
| Reference:
|
[4] Markushevich A.I.: Theory of Functions of a Complex Variable, III.Prentice Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0215964 |
| Reference:
|
[5] Nautiyal A., Rizvi S.R.H., Kapoor G.P.: On the approximation of an entire function in $L^2$-norm.Bull. Malaysian Math. Soc. (2) 5 (1982), 21-31. MR 0683808 |
| Reference:
|
[6] Reddy A.R.: Approximation of an entire function.J. Approx. Theory 3 (1970), 128-137. Zbl 0207.07302, MR 0259453 |
| Reference:
|
[7] Reddy A.R.: Best polynomial approximation to certain entire functions.J. Approx. Theory 5 (1972), 97-112. Zbl 0225.41006, MR 0404635 |
| Reference:
|
[8] 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. 88 (1970), 291-301. |
| Reference:
|
[9] Shah S.M.: Polynomial approximation of an entire function and generalized orders.J. Approx. Theory 19 (1977), 315-324. Zbl 0311.30034, MR 0440254 |
| Reference:
|
[10] Smirnov V.I., Lebedev N.A.: Functions of a Complex Variable: Constructive Theory.M.I.T. Press, Mass., USA, 1968. Zbl 0164.37503, MR 0229803 |
| Reference:
|
[11] Varga R.S.: On an extension of a result of Bernstein.J. Approx. Theory 1 (1968), 176-179. MR 0240528 |
| Reference:
|
[12] Winiarski T.: Approximation and interpolation of entire functions.Ann. Polon. MAth. 23 (1970), 259-273. Zbl 0205.37905, MR 0273032 |
| . |
