On one set of orthogonal harmonic polynomials

@inproceedings{Karachik1998OnOS,
  title={On one set of orthogonal harmonic polynomials},
  author={V. Karachik},
  year={1998}
}
A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others. Introduction For the investigation of harmonic polynomials a scalar product for homogeneous polynomials of degree m in the form 〈Pm(x), Qm(x)〉 = Pm(D)Qm(x) was introduced in [1]—one of the basic works on harmonic analysis— where the operator Pm(D) is obtained from the polynomial Pm(x) by replacing each variable xi on… Expand
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References

SHOWING 1-2 OF 2 REFERENCES
A Basis for Polynomial Solutions to Systems of Linear Constant Coefficient PDE's
Abstract LetKrepresent either the real or the complex numbers. LetPk,k=1, 2, …, rbe constant coefficient (with coefficients fromk) polynomials innvariables and letN⩽M={u(x)∈K[x]⩽M∣Pk(∂/∂x1, …, ∂/∂xn)Expand
Introduction to Fourier Analysis on Euclidean Spaces.
The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the actionExpand