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Some problems of real and stochastic analysis arising from applications quasi-periodic solutions of Hamiltonian evolution equations scaling limits for lattice gas models multivariate distributions with Gaussian conditional structure the minimal projection from L1 onto Pi-n proofs and proofs of the Eckart-Young theorem an analytic semigroup associated to a degenerate evolution equation degenerate nonlinear parabolic problems - the influence of probability theory an application of measure theory to perfect competition dilations of Hilbert-Schmidt class operator-valued measures and applications transient solution of the M/M/1 queueing system via randomization a characterization of Hida measure new results in the simplex method in linear programming an estimate of the semi stable measure of small balls in Banach spaces nonsquare constants of Orlicz spaces recursive multiple Wiener expansion for nonlinear filtering of diffusion processes a Berry-Esseen type estimate for Hilbert space valued U-statistics and on Bootstrapping Von Mises statistics on the strong form of the Faber theorem nonlinear filtering theory for stochastic reaction-diffusion equations an operator characterization of oscillatory harmonizable processes operator algebraic aspects for sufficiency nonlinear parabolic equations, Favard classes and regularity.

Motivated by the excellent work of Bill Davis and Jerry Uhl’s Differential Equations & Mathematica [1], we present in detail several little-known applications of the fast discrete Fourier transform (DFT), also known as FFT. Namely, we first examine the use of the FFT in multiplying univariate polynomials and integers and approximating polynomials with sines and cosines (also known as the fast Fourier fit or FFF). We then examine the use of the FFF in solving differential equations with Laplace transforms and rediscovering trigonometric identities.

Throughout the academic year 1986-87, the University of Illinois was host to a symposium on mathematical analysis which was attended by some of the leading figures in the field. This book arises out of this special year and lays emphasis on the synthesis of modern and classical analysis at the current frontiers of knowledge. The contributed articles by the participants cover the gamut of mainstream topics. This book will be essential to researchers in mathematical analysis.

Publisher Summary This chapter provides an overview of differentiation in Banach spaces. A Banach space X has a boundedly complete basis (Xn) if each x ∊ X has the unique expansion. If X has a boundedly complete basis, then every Lipschitz is differentiable almost everywhere. Since Lipschitz functions have separable ranges, it is clear that a Banach space has Radon-Nikodym property (RNP) if and only if each of its separable subspaces has RNP. But each separable subspace of a reflexive space is a separable dual space and hence has the RNP.