The paper is concerned with the introduction and study of multiresolution analysis based on the up function, which is an infinitely differentiable function supported on [0, 2]. Such analysis is, necessarily, nonstationary. It is shown that the approximation orders associated with the corresponding spaces are spectral, thus making the spaces attractive for the approximation of very smooth functi...

New continuous wavelets of compact support are introduced, which are related to the beta distribution. They can be built from probability distributions using ”blur”derivatives. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets...

In this paper, we obtain symmetric C∞ real-valued tight wavelet frames in L2(R) with compact support and the spectral frame approximation order. Furthermore, we present a family of symmetric compactly supported C∞ orthonormal complex wavelets in L2(R). A complete analysis of nonstationary tight wavelet frames and orthonormal wavelet bases in L2(R) is given.

A compactly supported radially symmetric function Φ : R → R is said to have Sobolev regularity k if there exist constants B ≥ A > 0 such that the Fourier transform of Φ satisfies A(1 + ‖ω‖) ≤ b Φ(ω) ≤ B(1 + ‖ω‖), ω ∈ R. Such functions are useful in radial basis function methods because the resulting native space will correspond to the Sobolev space W k 2 (R). For even dimensions d and integers ...

Radial Basis Functions (RBFs) are widely used in science, engineering and finance for constructing nonlinear models of observed data. Most applications employ activation functions from a relatively small list, including Gaussians, multi-quadrics and thin plate splines. We introduce several new candidate compactly supported RBFs for approximating functions in L (R) via overdetermined least squar...

We show that compactly supported wavelets in L (R) of scale N may be effectively parameterized with a finite set of spin vectors in C , and conversely that every set of spin vectors corresponds to a wavelet. The characterization is given in terms of irreducible representations of orthogonality relations defined from multiresolution wavelet filters.

The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L2(R) from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of “mother wavelets”, however, increases with the increase of the required smoothness. Two bivariate ...

The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L2(R) from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of “mother wavelets”, however, increases with the increase of the required smoothness. Two bivariate ...

Compact support is undoubtedly one of the wavelet properties that is given the greatest weight both in theory and applications. It is usually believed to be essential for two main reasons : (1) to have fast numerical algorithms, and (2) to have good time or space localization properties. Here, we argue that this constraint is unnecessarily restrictive and that fast algorithms and good localizat...

Let • • • C K_ ] c Vq c Vx c • • • be a multiresolution analysis of L2 generated by the mth order 5-spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces .quently, the two finite sequences that describe the two-scale relations of Nm(x) and i//m(x) in terms of Nm(2x-j), ;6Z, yield an efficie...