Abstract
We study the orthogonal projection of homogeneous polynomials onto the space of homogeneous polyharmonic polynomials. To do this we derive the decomposition of homogeneous polynomials in terms of the Kelvin transform of derivatives of the fundamental solution \(|x|^{2-n}\) or \(\log |x|\). We consider also the vector bases of the space of homogeneous polyharmonic polynomials and study the problem of convergence of orthogonal series.
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References
N. Aronszajn, T. M. Creese and L. J. Lipkin, Polyharmonic Functions, Clarendon Press (Oxford, 1983).
S. Axler and W. Ramey, Harmonic polynomials and Dirichlet-type problems, Proc. Amer. Math. Soc., 123 (1995), 3765–3773.
S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd ed., Springer- Verlag (New York, 2001).
M. Brelot and G. Choquet, Polynômes harmoniques et polyharmoniques, in: Second colloque sur les équations aux dérivées partielles, Bruxelles, 1954, Georges Thone, Masson & Cie, (Lig̀e, Paris, 1955), pp. 45–66.
M. D. Buhmann, Radial Basis Functions. Theory and Implementations, Cambridge University Press (Cambridge, 2003).
A. Fryant and M. K. Vemuri, Pythagorean identity for polyharmonic polynomials, Int. J. Math. Math. Sci., 29 (2002), 115–119.
F. Gazzola, H. Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag (New York, 2010).
H. Grzebuła and S. Michalik, A Dirichlet type problem for complex polyharmonic functions, Acta Math. Hungar., 153 (2017), 216–229.
H. Grzebuła and S. Michalik, Spherical polyharmonics and Poisson kernels for polyharmonic functions, Complex Var. Elliptic Equ., 64 (2019), 420–442.
O. Kounchev, Multivariate Polysplines: Applications to Numerical and Wavelet Analysis, Academic Press (San Diego, 2001).
E. Lundberg and H. Render, The Khavinson–Shapiro conjecture and polynomial decompositions, J. Math. Anal. Appl., 376 (2011), 506–513.
G. Łysik, On the mean-value property for polyharmonic functions, Acta Math. Hungar., 133 (2011), 133–139.
W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines, J. Approx. Theory, 60 (1990), 141–156.
S. Michalik, Summable solutions of some partial differential equations and generalised integral means, J. Math. Anal. Appl., 444 (2016), 1242–1259.
H. Render, A characterization of the Khavinson–Shapiro conjecture via Fischer operators, Potential Anal., 45 (2016), 539–543.
H. Render, Cauchy, Goursat and Dirichlet problems for Holomorphic partial differential equations, Comput. Methods Funct. Theory, 10 (2011), 519–554.
H. Render, Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions, Duke Math. J., 142 (2008), 313–352.
H. Render, Reproducing kernels for polyharmonic polynomials, Arch. Math., 91 (2008), 136–144.
H. S. Shapiro, An algebraic theorem of E. Fischer, and the holomorphic Goursat problem, Bull. London Math. Soc., 21 (1989), 513–537.
J. Siciak, Holomorphic continuation of harmonic functions, Ann. Polon. Math., 29 (1974), 67–73.
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The authors are grateful to the anonymous referees for the valuable comments and suggestions to improve the paper.
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Grzebuła, H., Michalik, S. On polyharmonic polynomials. Acta Math. Hungar. 169, 325–348 (2023). https://doi.org/10.1007/s10474-023-01323-9
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DOI: https://doi.org/10.1007/s10474-023-01323-9