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Approximation and interpolation of generalized biaxisymmetric potentials. (English) Zbl 0957.30028

From the introduction: “Let \(F^{\alpha,\beta}\) be a real-valued regular solution to the generalized biaxisymmetric potential equation \[ \left(\frac {\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+\frac{2\alpha+1} {x}\frac{\partial} {\partial x}+\frac {2\beta+1} {y} \frac {\partial} {\partial y}\right)F^{\alpha, \beta}=0, \quad\alpha >\beta>-\frac 12, \] where \((\alpha,\beta)\) are fixed in a neighbourhood of the origin and the analytic Cauchy condition \(F^{\alpha,\beta}_x(0,y)-F^{\alpha, \beta}_y(x,0)=0\) is satisfied along the singular lines in the open hypersphere \(\Sigma ^{\alpha,\beta} _r:x^2+y^2<r^2\). Such functions with even harmonic functions are referred to as generalized biaxisymmetric potentials (GBASP) having local expansions of the form \[ F^{ \alpha, \beta} (x,y)=\sum^\infty_{n=0} a_nR^{\alpha,\beta} _n(x, y) \] such that \[ R^{\alpha, \beta}_n(x,y)= (x^2+y^2)^nP^{\alpha, \beta}_n((x^2-y^2)/ (x^2+y^2))/P^{\alpha,\beta}_n(1), \quad n= 0,1,2, \cdots, \] where the \(P^{\alpha, \beta}_n\) are Jacobi polynomials [R. Askey, Orthogonal polynomials and special functions (1975; Zbl 0298.33008); G. Szegő, Orthogonal polynomials (1959; Zbl 0089.27501)]. The results of P. A. McCoy [J. Approximation Theory 25, No. 2, 153-168 (1979; Zbl 0462.41002); and O. P. Juneja, G. P. Kapoor and S. K. Bajpai [J. Reine Angew. Math. 282, 53-67 (1976; Zbl 0321.30031); J. Reine Angew. Math. 290, 180-190 (1977; Zbl 0501.30021)] are extended. In particular, the concepts of proximate order and \((p,q)\)-scale of Juneja are used for studying a wider class of solutions \(F^{\alpha,\beta}\) of the above equation.

MSC:

30E10 Approximation in the complex plane
41A10 Approximation by polynomials
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