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- Bernd Fischer, Jürgen Prestin
- Math. Comput.
- 1997

We present a unified approach for the construction of polynomial wavelets. Our main tool are orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.

- H. N. Mhaskar, J. Prestin
- 2000

We propose a class of algebraic polynomial frames, which are computationally easier to implement than polynomial bases. We also discuss the weighted L p-stability of our frames for 1 ≤ p ≤ ∞. Our analysis is based on orthogonal polynomials with respect to the weight in question, but the frame bounds are independent of the system of orthogonal polynomials… (More)

- Hrushikesh Narhar Mhaskar, Jürgen Prestin
- Adv. Comput. Math.
- 2000

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)

- Daniel Potts, Jürgen Prestin, Antje Vollrath
- Numerical Algorithms
- 2009

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)

- Frank Filbir, Hrushikesh Narhar Mhaskar, Jürgen Prestin
- Journal of Approximation Theory
- 2009

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)

- H. N. Mhaskar, J. Prestin
- 1999

Let f be a piecewise analytic function on the unit interval (respectively , the unit circle of the complex plane). Starting from the Chebyshev (respectively, Fourier) coefficients of f , we construct a sequence of fast decreasing polynomials (respectively, trigonometric polynomials) which " detect " the points where f fails to be analytic, provided f is not… (More)

In this paper we present an algorithm for the fast Fourier transform on the rotation group SO(3) which is based on the fast Fourier transform for nonequispaced nodes on the three-dimensional torus. This algorithm allows to evaluate the SO(3) Fourier transform of B-band-limited functions at M arbitrary input nodes in O(M + B 3 log 2 B) flops instead of O(M B… (More)

- Jürgen Prestin, Ewald Quak
- Numerical Algorithms
- 1995

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)

- Hrushikesh Narhar Mhaskar, Francis J. Narcowich, Jürgen Prestin, Joseph D. Ward
- Adv. Comput. Math.
- 2000

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)