Polynomial Harmonic Decompositions

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For real polynomials in two indeterminates a classical polynomial harmonic decomposition (cf. (1) below) is extended from square-norm divisors to conic ones. The main result is then applied to obtain a full polynomial harmonic decomposition, and to solve a Dirichlet problem with polynomial boundary data.

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  • [1] S. Axler P. Bourdon W. Ramey Harmonic Function Theory Graduate Texts Math. 137 Springer New-York 1992.

  • [2] S. Axler W. Ramey Harmonic Polynomials and Dirichlet-Type Problems Proc. Amer. Math. Soc. 123 (1995) 3765-3773.

  • [3] W. Conley Complex Analysis Birkhäuser Boston 2004.

  • [4] F. Colombo I. Sabadini F. Sommen and D. Struppa Analysis of Dirac Systems and Computational Algebra Birkhäuser Boston 2004.

  • [5] B. Reznick Homogeneous Polynomial Solutions to Constant Coefficient PDE's Adv. Math. 117 (1996) 179-192.

  • [6] M. Shubin Pseudodifferential Operators and Spectral Theory Springer NewYork 1987.

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