Tensor–tensor products with invertible linear transforms

Invertible Linear Transforms Implemented by Integer Mapping

Abstract The general integer mapping theory of invertible linear transforms is considered in this paper. Two terms of integer factor and elementary triangular matrix are introduced. We prove that there exists an integer implementation if a linear transform is invertible and in finite dimensional space. The integer implementation of an invertible linear transform has several advantages such as (...

Prediction in invertible linear processes

We construct root-n consistent plug-in estimators for conditional expectations of the form E(h(Xn+1, . . . , Xn+m)|X1, . . . , Xn) in invertible linear processes. More specifically, we prove a Bahadur type representation for such estimators, uniformly over certain classes of not necessarily bounded functions h. We obtain in particular a uniformly root-n consistent estimator for the m-dimensiona...

Inverse Transforms of Products of Legendre Transforms

is obtained by applying successive integration by parts to the integral represented by the left-hand member of this equation so that the new integrand is the product of F by [(1 —x2)P„' ]', and by replacing this second factor by — n(n + l)Pn according to Legendre's differential equation [l]. Thus certain boundary value problems in ordinary and partial differential equations that involve the dif...

Efficient Estimation in Invertible Linear Processes

Fast linear canonical transforms.

The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande-Tukey fast Fourier tran...