Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere. (English) Zbl 1533.65038
It has been known for many years that many real-world applications can be modeled as spherical problems. A critical task of spherical modeling is to find an effective data fitting strategy to approximate the underlying mapping between input and output data. Hyperinterpolation, introduced by Ian Sloan is an example of a method for fitting spherical data. More precisely, Hyperinterpolation of degree \(n\) is a discrete approximation of the \(L_2\)-orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most \(2n\).
The paper under review, studies the approximation of continuous functions on the unit sphere by spherical polynomials of degree \(n\) via hyperinterpolation where the hyperinterpolation is constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. The paper provides an approximation error estimate consisting of two terms, the first representing the error estimate of the original hyperinterpolation of full quadrature exactness and the second, a compensation term for the loss of exactness degrees. A method and example to controlling the new term is provided.
The paper is well written with a good set of references.
The paper under review, studies the approximation of continuous functions on the unit sphere by spherical polynomials of degree \(n\) via hyperinterpolation where the hyperinterpolation is constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. The paper provides an approximation error estimate consisting of two terms, the first representing the error estimate of the original hyperinterpolation of full quadrature exactness and the second, a compensation term for the loss of exactness degrees. A method and example to controlling the new term is provided.
The paper is well written with a good set of references.
Reviewer: Steven B. Damelin (Ann Arbor)
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 41A10 | Approximation by polynomials |
| 41A55 | Approximate quadratures |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 33C55 | Spherical harmonics |
Keywords:
hyperinterpolation; quadrature; exactness; Marcinkiewicz-Zygmund inequality; spherical \(t\)-designs; QMC designsReferences:
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