Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let A∈Rd×d, d≥1 be a dilation matrix with integer entries and |detA|=2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is 2d+d and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case d=2 and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.The first author was partially supported by MEC/MICINN Grant #MTM2011-27998 (Spain)

Variational-Wavelet Approach to RMS Envelope Equations

We present applications of variational-wavelet approach to nonlinear (rational) rms envelope equations. We have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic beam motion and arbitrary variable coefficients. Also we consider more flexible variational method which is based on biorthogonal wavelet approach.Comment: 21 pages, 8 figures, LaTeX2e, presented at Second ICFA Advanced Accelerator Workshop, UCLA, November, 199

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of $N$ samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to $J$, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App

Frames, semi-frames, and Hilbert scales

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.Comment: 27 pages; Numerical Functional Analysis and Optimization, 33 (2012) in press. arXiv admin note: substantial text overlap with arXiv:1101.285