Kasana, Harvir S.; Kumar, Devendra \(L^p\)-approximation of generalized biaxially symmetric potentials over Carathéodory domains. (English) Zbl 1150.41357 Math. Slovaca 55, No. 5, 563-572 (2005). Summary: Let \(F^{\alpha ,\beta }\) be a real generalized biaxially symmetric potentials (GBASP) defined on the Carathéodory domain and let \(L^\rho (D)\) be the class of functions \(F^{\alpha ,\beta }\) holomorphic in \(D\) such that \(\| F^{\alpha ,\beta }\| _{D,p} =\left (A^{-1}\iint \limits _D| F^{\alpha ,\beta }| \operatorname {d}\!x\,{\operatorname {d}}y\right )^{1/p}\), \(A\) is the area of the domain \(D\). For \(F^{\alpha ,\beta }\in L^p(D)\), set \(E^p_n\bigl (F^{\alpha ,\beta }\bigr ) ={\text{inf}}\{\| F^{\alpha ,\beta } - P^{\alpha ,\beta }\| _{D,p} \:P^{\alpha ,\beta }\in H_n\}\), \(H_n\) consists of all even biaxially symmetric harmonic polynomials of degree at most 2\(n\). This paper deals with the growth of entire function GBASP in term of approximation error in \(L^p\)-norm on \(D\). The analysis utilizes the Bergram and Gilbert integral operator method to extend result from clasical function theory on the best polynomial approximation of analytic functions of a complex variable. Cited in 1 Document MSC: 41A80 Remainders in approximation formulas 35J15 Second-order elliptic equations 30E10 Approximation in the complex plane 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions Keywords:generalized biaxially symmetric potential; \(L^p\)-norm; approximation error; generalized growth parameter; Fourier coefficient PDFBibTeX XMLCite \textit{H. S. Kasana} and \textit{D. Kumar}, Math. Slovaca 55, No. 5, 563--572 (2005; Zbl 1150.41357) Full Text: EuDML References: [1] ASKEY R.: Orthogonal Polynomials and Special Functions. Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, PA, 1975. · Zbl 0298.26010 [2] BERNSTEIN S. N.: Lecons sur les Properties Extremales et la Meilleure Approximation des Fonctions Analytiques d’une Variable Reele. Gauthier-Villars, Paris, 1926. [3] GILBERT R. P.: Function Theoretic Methods in Partial Differential Equations. Math. Sci. Engrg. 54, Academic Press, New York, 1969. · Zbl 0187.35303 [4] GILBERT R. P.-NEWTON R. G.: Analytic Methods in Mathematical Physics. Gordon and Breach Sci. Publ., New York, 1970. · Zbl 0194.29903 [5] KAPOOR G. P.-NAUTIYAL A.: Polynomial approximation of an entire function of slow growth. J. Approx. Theory 32 (1981), 64-75. · Zbl 0495.41005 [6] KASANA H. S.-KUMAR D.: On approximation and interpolation of entire functions with index-pair \((p,q)\). Publ. Mat. 38 (1994), 255 267. · Zbl 0831.30023 [7] KUMAR D.-KASANA H. S.: On approximation of entire functions over Carathedory domains. Comment. Math. Univ. Carolin. 35 (1994), 681-689. · Zbl 0815.30019 [8] MARKUSHEVIC A. I.: Theory of Functions of a Complex Variables. Prentice Hall, Inc. Englewood Cliffs, NJ, 1967. [9] MCCOY P. A.: Polynomial approximation of generalized biaxisymmetric potentials. J. Approx. Theory 25 (1979), 153-168. · Zbl 0462.41002 [10] MCCOY P. A.: Best \(L^p\)-approximation of generalized biaxisymmetric potentials. Proc. Amer. Math. Soc. 79 (1980), 435-440. · Zbl 0451.31014 [11] SMIRNOV V. I.-LEBEDEV N. A.: Functions of a Complex Variable: Constructive Function Theory. MIT Press, Mass USA, 1968. · Zbl 0164.37503 [12] SZEGÖ G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. 22, Amer. Math. Soc, Providence, RI, 1967. · JFM 65.0278.03 [13] WINIARSKI T. N.: Approximation and interpolation of entire functions. Ann. Polon. Math. 23 (1970), 259-273. · Zbl 0205.37905 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.