On Filtered Polynomial Approximation on the Sphere

Abstract

Localised polynomial approximations on the sphere have a variety of applications in areas such as signal processing, geomathematics and cosmology. Filtering is a simple and effective way of constructing a localised polynomial approximation. In this thesis we investigate the localisation properties of filtered polynomial approximations on the sphere. Using filtered polynomial kernels and a special numerical integration (quadrature) rule we construct a fully discrete need let approximation. The localisation of the filtered approximation can be seen from the localisation properties of its convolution kernel. We investigate the localisation of the filtered Jacobi kernel, which includes the convolution kernel for filtered approximation on the sphere as a particular example. We prove the precise relation between the filter smoothness and the decay rate of the corresponding filte red Jacobi kernel over local and global regions. The difference in localisation properties between Fourier and filtered approximations can be illustrated by their Riemann localisation. We show that the Riemann localisation property holds for the Fourier-Laplace partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimensions. We then prove that the filtered approximation with sufficiently smooth filter has the Riemann localisation property for spheres of any dimensions. Filtered convolution kernels with a special filter become spherical need lets, which are highly localised zonal polynomials on the sphere with centres at the nodes of a suitable quadrature rule. The original semidiscrete spherical needlet approximation has coefficients defined by inner product integrals. We use an appropriate quadrature rule to construct a fully discrete version. We prove that the fully discrete spherical need let approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace partial sum with inner products replaced by appropriate quadrature sums. From this we establish error bounds for the fully discrete need let approximation of functions in Sobolev spaces on the sphere. The power of the need let approximation for local approximation is shown by numerical experiments that use low-level need lets globally together with high-level need lets in a local region. Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media , now or here after known , subject to the provisions of the Copyright Act 1968. I retain all property rights , such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation.

Distributed Learning via Filtered Hyperinterpolation on Manifolds

“…We obtain an error bound O N −r /d for the noiseless setting on general manifolds (see Theorem 6). This result generalizes the same bound that was previously obtained on the sphere [35]. Since the bound on the sphere is optimal, the new bound is also optimal.…”

“…From the perspective of information-based complexity it is interesting to observe that if the target function f * is in the Sobolev space W s p (M), s > 0, the convergence rate is optimal in the sense of optimal recovery. This is due to the fact that on a real unit sphere when one uses optimal-order number of points 10) is optimal, as proved by Wang and Sloan [35] and Wang and Wang [36]. Theorem 6 can be viewed as the non-distributed filtered hyperinterpolation for clean data, where the estimator uses the whole data set in one machine.…”

“…The proof of optimal-order error for filtered hyperinterpolation utilizes its decomposition by framelets on manifolds [35,38,43]. Given H ∈ C κ (R + ), κ ≥ 1, we define recursively the contributions of levels j ∈ N 0 for f ∈ L p (M) by…”

“…Then, with (35), (34) and cn d+τ ≤ |D * | ≤ c n 2d , τ ∈ (0, d], M) , where c 5 and c depend only on μ(M), c 1 , d, r , and the filter H and its smoothness κ, and C from Lemma 6.…”

Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

“…Wang and Sloan investigated the corresponding problem on the sphere, and gave the compact condition on η for which the operator norms of the filtered polynomial operators V S L,η on the sphere are uniformly bounded. Following the way in [38], Li obtained in [22] that the operator norms V L,η are uniformly bounded whenever η ∈ W ⌊ d+2µ+1…”