Matrix reenement equations are functional equations of the form f(x) = P N k=0 c k f(2x ? k), where the coeecients c k are matrices and f is a vector-valued function. Reenement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar reenement equations (where the coeecients c k are scalars) are known. This paper considers analogous ques...

Quantum time correlation functions obtained from the linear response theory are defined in terms of a scalar product involving the Kubo transform. This form of the scalar product also insures that the nonequilibrium average of the random force fluctuations corresponding to an observable vanish so that the macroscopic law is obtained on long distance and time scales and the Onsager regression hy...

We present a detailed motivation for the notions of the chromatic derivatives and the chromatic expansions. The chromatic derivatives are special, numerically robust linear differential operators. The chromatic expansions are the associated local expansions, which possess features of both the Taylor and the Nyquist expansions. We give simplified treatment of some of the basic properties of the ...

A scalar valued set function on a Cartesian product of σ-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequali...

We investigate the possibilities for attaining two Levenshtein upper bounds for spherical codes. We find the distance distributions of all codes meeting these bounds. Then we show that the fourth Levenshtein bound can be attained in some very special cases only. We prove that no codes with an irrational maximal scalar product meet the third Levenshtein bound. So in dimensions 3 ≤ n ≤ 100 exactl...