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We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either nitely many Fourier coeecients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also(More)
In this paper we present algorithms to calculate the fast Fourier synthesis and its adjoint on the rotation group SO(3) for arbitrary sampling sets. They are based on the fast Fourier transform for nonequispaced nodes on the three-dimensional torus. Our algorithms evaluate the SO(3) Fourier synthesis and its adjoint, respectively, of B-bandlimited functions(More)
The purpose of this paper is to establish L p error estimates, a Bernstein inequality , and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L p Bessel-potential Sobolev norms of functions in this space in terms of the minimal(More)
Let α, β ≥ −1/2, and for k = 0, 1, · · ·, pk (α,β) denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H, α, and β such that ˛ ˛ ˛ ˛ ˛ ∞ X k=0 Hk,npk (α,β) (cos θ)pk (α,β) (cos ϕ) ˛ ˛ ˛ ˛ ˛ ≤ c1n 2 max(α,β)+2 exp(−cn(θ − ϕ) Specializing to the(More)
The aim of this paper is to describe explicit decomposition and reconstruction algorithms for nested spaces of trigonometric polynomials. The scaling functions of these spaces are defined as fundamental polynomials of Lagrange interpolation. The interpolatory conditions and the construction of dual functions are crucial for the approach presented in this(More)
We introduce a class of polynomial frames suitable for analyzing data on the surface of the unit sphere of a Euclidean space. Our frames consist of polynomials, but are well localized, and are stable with respect to all the L p norms. The frames belonging to higher and higher scale wavelet spaces have more and more vanishing moments. 1 to reproduce and(More)