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Fourier orthogonal series on a paraboloid

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Abstract

We study the orthogonal structure and Fourier orthogonal series on the surface of a paraboloid

$$\mathbb{V}_{0}^{d+1}=\{(x,t):\Vert x\Vert=\sqrt{t},x\in \mathbb{R}^{d},0\le t<1\}.$$

The reproducing kernels of the orthogonal polynomials with respect to tβ(1 − t)γ on \(\mathbb{V}_{0}^{d+1}\) are related to the reproducing kernels of the Jacobi polynomials on the parabolic domain {(x1, x2): x 21 ≤ x2 ≤ 1} in ℝ2. This connection serves as an essential tool for our study of the Fourier orthogonal series on the surface of the paraboloid, which allow us, in particular, to study the convergence of the Cesàro means on the surface. Analogous results are also established for the solid paraboloid bounded by \(\mathbb{V}_{0}^{d+1}\) and the hyperplane t =1.

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Acknowledgement

The author thanks an anonymous referee for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, Oregon, 97403-1222, USA

    Yuan Xu

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  1. Yuan Xu
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Correspondence to Yuan Xu.

Additional information

The author is partially supported by Simons Foundation Grant #849676.

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Xu, Y. Fourier orthogonal series on a paraboloid. JAMA 149, 251–279 (2023). https://doi.org/10.1007/s11854-022-0251-2

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  • DOI: https://doi.org/10.1007/s11854-022-0251-2

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