A nonlinear generalization of the S-transform known in Gaussian analysis is introduced. It is deened as a nonlinear operator on topolog-ical spaces of entire functions of innnitely many variables with growth of at most second order and nite type restricted to a certain value. This nonlinear S-transform (NLST) has xed points a certain family of which is found and analyzed. In particular, their s...

We introduce the operator Z that is often called the Zak transform. Our definition is a bit different from the one that usually appears in the literature. We will discuss this difference and will also give a historical account that the reader may find particularly interesting. In order to do this, however, we need to present our treatment of the operator Z (and Z̃) which shows that the Fourier t...

The S–transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B–valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra B. Also, a new proof of the additivity of the R–transform in this setting is given.

In 1983 L. G. Brown introduced a spectral distribution measure for non-normal elements in a nite von Neumann algebra M with respect to a xed normal faithful tracial state. In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T = UH where U is a Haar unitary and U and H are-free. (When Ker T = f0g this is equivalent to that (T; T) is an R-diagonal pa...

The algebra Mul[[B]] of formal multilinear function series over an algebra B and its quotient SymMul[[B]] are introduced, as well as corresponding operations of formal composition, which are associative. In the setting of Mul[[B]], the unsymmetrized R– and T–transforms of random variables in B–valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usu...

Generalization of the well-known Walsh–Hadamard transform (WHT), namely center-weighted Hadamard transform (CWHT) and complex reverse-jacket transform (CRJT) have been proposed and their fast implementation and simple index generation algorithms have recently been reported. These transforms are of size 2 2 for integral values or , and defined in terms of binary radix representation of integers....

A lattice structure based on the singular value decomposition (SVD) is introduced. The lattice can also be proven to use a minimal number of delay elements and to completely span a large class of M-channel linear phase perfect reconstruction filter bank (LPPRFB): all analysis and synthesis filters have the same FIR length of L = K M , sharing the same center of symmetry. The lattice also struct...

The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N + M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with O(q) distinct Gaus...