A Multivariate Fast Discrete Walsh Transform with an Application to Function Interpolation

For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $O(N \log N)$ operations, where $N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented

Almost everywhere convergence of Fej\'er means of two-dimensional triangular Walsh-Fourier series

In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fej\'er (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence $\sigma_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n-1}S_{k, n-k}f\to f$ ($n\to\infty$) for each integrable two-variable function $f$

Cross talk in phase-coded holographic memories

The cross talk between holograms multiplexed with Walsh-Hadamard phase codes is analyzed. Each hologram is stored with a reference beam that consists of N phase-coded plane waves. The signal-to-noise ratio (SNR) is calculated for a recording schedule for which the center of each stored image coincides with the nulls of the selectivity function for the adjacent plane-wave components of the reference beam. The SNR characteristics for phase coding with Walsh-Hadamard phase codes are then compared with the SNR for angle and wavelength multiplexing

On the universal function for weighted spaces L^p_u[0,1], p>=1

In the paper it is shown that there exist a function g from L1[0,1] and a weight function 0<u(x)<=1, so that g is universal for each classes L^p_u[0,1], p>= 1 with respect to signs-subseries of its Fourier-Walsh series

Walsh function generator for the Electronically Scanned Thinned Array Radiometer (ESTAR) instrument

A prototype Walsh Function Generator (WFG) for the ESTAR (Electronically Scanned Thinned Array Radiometer) instrument has been designed and tested. Implemented in a single Xilinx XC3020PC68-50 Field Programmable Gate Array (FPGA), it generates a user-programmable set of 32 consecutive Walsh Functions for noise cancellation in the analog circuitry of the Front-End Modules (FEM's). It is implemented in a 68-pin plastic leaded chip carrier (PLCC) package, is fully testable, and can be used for noise cancellation periods as small as 2 msec