On the approximation of entire functions over Carathéodory domains
Kumar, Devendra ; Kasana, Harvir S.
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994), p. 681-689 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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 $D$.

Publié le : 1994-01-01
Classification:  30D15,  30E10 
@article{118709,
     author = {Devendra Kumar and Harvir S. Kasana},
     title = {On the approximation of entire functions over Carath\'eodory domains},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {35},
     year = {1994},
     pages = {681-689},
     zbl = {0815.30019},
     mrnumber = {1321238},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118709}
}

Kumar, Devendra; Kasana, Harvir S. On the approximation of entire functions over Carathéodory domains. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 681-689. http://gdmltest.u-ga.fr/item/118709/

Bernstein S.N. Constructive theory of functions (1905-1930) in collected works, I, Akademia Nauk SSSR, 1952. | MR 0048360

Juneja O.P. Approximation of an entire function, J. Approx. Theory 11 (1974), 343-349. (1974) | MR 0390608 | Zbl 0285.30030

Kapoor G.P.; Nautiyal A. Polynomial approximation of an entire function of slow growth, J. Approx. Theory 32 (1981), 64-75. (1981) | MR 0629582 | Zbl 0495.41005

Markushevich A.I. Theory of Functions of a Complex Variable, III, Prentice Hall, Inc., Englewood Cliffs, N.J., 1967. | MR 0215964

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. (1982) | MR 0683808

Reddy A.R. Approximation of an entire function, J. Approx. Theory 3 (1970), 128-137. (1970) | MR 0259453 | Zbl 0207.07302

Reddy A.R. Best polynomial approximation to certain entire functions, J. Approx. Theory 5 (1972), 97-112. (1972) | MR 0404635 | Zbl 0225.41006

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. (1970)

Shah S.M. Polynomial approximation of an entire function and generalized orders, J. Approx. Theory 19 (1977), 315-324. (1977) | MR 0440254 | Zbl 0311.30034

Smirnov V.I.; Lebedev N.A. Functions of a Complex Variable: Constructive Theory, M.I.T. Press, Mass., USA, 1968. | MR 0229803 | Zbl 0164.37503

Varga R.S. On an extension of a result of Bernstein, J. Approx. Theory 1 (1968), 176-179. (1968) | MR 0240528

Winiarski T. Approximation and interpolation of entire functions, Ann. Polon. MAth. 23 (1970), 259-273. (1970) | MR 0273032 | Zbl 0205.37905