I describe the work of Olevski˘ ı, Tao and others on the rearrangement of orthogonal series. It turns out that arbitrary rearrangements produce trouble for all orthogonal and wavelet methods, that decreasing rearrangements produce trouble for Fourier series, but that wavelet expansions continue to work well under decreasing rearrangement.

h a series of orthogonal polynomials. This representation corresponds physically to a system conFig. VIII-1. Nonlinear filter. taining only multipliers as nonlinear elements. The suggestion offered in this report is to represent f by an orthogonal trigonometric series that corresponds physically to a system containing phase modulators. To simplify matters, let us first assume that the signals x...

In this paper, using both an analytic and algorithmic approach, we derive the coefficients Dm(n, a) of the multiplication formula pn(ax) = n ∑ m=0 Dm(n, a)pm(x) or the translation formula pn(x +a) = n ∑ m=0 Dm(n, a)pm(x), where {pn}n≥0 is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the q-classical orthog...

The ubiquity of strange attractors in nature suggests that nonlinear modeling techniques can improve performance in some signal processing applications. We introduce Mixed State Markov Models (MSMMs), a refinement of Hidden Filter HMMs, and apply both to a synthetic Double Scroll time series. Forecasts by HFHMMs diverge after a few steps. Using ad hoc procedures, forecasts by MSMMs, even models...

Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J-orthogonal) functions of discrete variables and may be interpreted as...

We derive closed formulas for the prices of European options andtheir sensitivities when the underlying asset follows a double-exponentialjump diffusion model, as considered by S. Kou in 2002. This author hasderived the option price by making use of double series where each termrequires the computation of a sequence of special functions, such thatthe implementation remains difficult for a large...