Joachim Stöckler

Error estimates for scattered data interpolation on spheres

We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

Affine frames, quasi-affine frames, and their duals

The notion of quasi-affine frame was recently introduced by Ron and Shen [9] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in [9]. Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems.

Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments

When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2=L2(R) with dilation integer factor M≥2, the standard “matrix extension” approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M=2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M=2 to arbitrary integer M≥2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M−1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M−1 such frame generators. Linear spline examples are given for M=3,4 to demonstrate our constructive approach.

Gabor frames and totally positive functions

Let g be a totally positive function of finite type, i.e., ĝ(ξ) = ∏M ν=1(1 + 2πiδνξ) −1 for δν ∈ R and M ≥ 2. Then the set {eg(t − αk) : k, l ∈ Z} is a frame for L(R), if and only if αβ < 1. This result is a first positive contribution to a conjecture of I. Daubechies from 1990. So far the complete characterization of lattice parameters α, β that generate a frame has been known for only six window functions g. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.Let

Analytic wavelets generated by radial functions

g

An identity for multivariate Bernstein polynomials

be a totally positive function of finite type. Then the Gabor set

Algorithms for cardinal interpolation using box splines and radial basis functions

\{e^{2\pi i \beta l t} g(t-\alpha k), k,l \in Z \}

New polynomial preserving operators on simplices: direct results

is a frame for

Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

L^2(R)

Discretized Gabor Frames of Totally Positive Functions

, if and only if