8 research outputs found

    Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

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    This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree nn via hyperinterpolation. Hyperinterpolation of degree nn is a discrete approximation of the L2L^2-orthogonal projection of degree nn with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2n2n. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.Comment: 22 pages, 7 figure
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