Best 𝐿^{𝑝}-approximation of generalized biaxisymmetric potentials

McCoy, Peter A.

doi:10.1090/S0002-9939-1980-0567987-5

Best $L^{p}$-approximation of generalized biaxisymmetric potentials
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Proc. Amer. Math. Soc. 79 (1980), 435-440 Request permission

Abstract:

Let F be a real-valued generalized biaxisymmetric potential (GBASP ) in ${L^p}(p \geqslant 1)$ on $\Sigma$, the open unit sphere about the origin. Convergence of a sequence of best harmonic polynomial approximates to F in ${L^p}$ identifies those F that harmonically continue as entire function GBASP and determines their order and type. The analysis utilizes the Bergman and Gilbert Integral Operator Method to extend results from classical function theory on the best polynomial approximation of analytic functions of one complex variable.

References

Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145, DOI 10.1137/1.9781611970470
Stefan Bergman, Integral operators in the theory of linear partial differential equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 23, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0141880, DOI 10.1007/978-3-642-64985-1

Leçon sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle

David L. Colton, Partial differential equations in the complex domain, Research Notes in Mathematics, Vol. 4, Pitman Publishing, London-San Francisco, Calif.-Melbourne, 1976. MR 0481427
Robert P. Gilbert, Function theoretic methods in partial differential equations, Mathematics in Science and Engineering, Vol. 54, Academic Press, New York-London, 1969. MR 0241789
Robert P. Gilbert, Constructive methods for elliptic equations, Lecture Notes in Mathematics, Vol. 365, Springer-Verlag, Berlin-New York, 1974. MR 0447784, DOI 10.1007/BFb0064475
Robert P. Gilbert and Roger G. Newton (eds.), Analytic methods in mathematical physics, Gordon and Breach Science Publishers, New York-London-Paris, 1970. Based on the Symposium held at Indiana University, Bloomington, Indiana, June 2–6, 1968. MR 0327404
I. I. Ibragimov and N. I. Šihaliev, Best polynomial approximation in a space of analytic functions, Dokl. Akad. Nauk SSSR 227 (1976), no. 2, 280–283 (Russian). MR 0422626

Foundations of potential theory

B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975, DOI 10.1090/mmono/005
Peter A. McCoy, Polynomial approximation and growth of generalized axisymmetric potentials, Canadian J. Math. 31 (1979), no. 1, 49–59. MR 518705, DOI 10.4153/CJM-1979-006-7
Peter A. McCoy, Polynomial approximation of generalized biaxisymmetric potentials, J. Approx. Theory 25 (1979), no. 2, 153–168. MR 531104, DOI 10.1016/0021-9045(79)90005-4
Peter A. McCoy, Approximation and harmonic continuation of axially symmetric potentials in $E^{3}$, Pacific J. Math. 81 (1979), no. 2, 481–491. MR 547615, DOI 10.2140/pjm.1979.81.481
A. R. Reddy, A contribution to best approximation in the $L^{2}$ norm, J. Approximation Theory 11 (1974), 110–117. MR 355066, DOI 10.1016/0021-9045(74)90022-7
A. R. Reddy, Approximation of an entire function, J. Approximation Theory 3 (1970), 128–137. MR 259453, DOI 10.1016/0021-9045(70)90020-1
E. B. Saff, Regions of meromorphy determined by the degree of best rational approximation, Proc. Amer. Math. Soc. 29 (1971), 30–38. MR 281930, DOI 10.1090/S0002-9939-1971-0281930-2

Orthogonal polynomials

Richard S. Varga, On an extension of a result of S. N. Bernstein, J. Approximation Theory 1 (1968), 176–179. MR 240528, DOI 10.1016/0021-9045(68)90020-8