we show that the character space of the vector-valued lipschitz algebra $lip^{alpha}(x, e)$ of order $alpha$ is homeomorphic to the cartesian product $xtimes m_e$ in the product topology, where $x$ is a compact metric space and $e$ is a unital commutative banach algebra. we also characterize the form of each character on $lip^{alpha}(x, e)$. by appealing to the injective tensor product, we then...

Let D = 1 or D be a fundamental discriminant [1]. The Kronecker-Jacobi-Legendre symbol (D/n) is a completely multiplicative function on the positive integers: µ D n ¶ = ⎧ ⎪ ⎨

Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the reproducing kernel Hilbert space and develop a method for learning prediction functions that accommodate such non-linearities. The method not only learns the p...

Let a a1, a2, . . . , am ∈ C be an m-dimensional vector. Then, it can be identified with an m ×m circulant matrix. By using the theory of matrix-valued wavelet analysis Walden and Serroukh, 2002 , we discuss the vector-valued multiresolution analysis. Also, we derive several different designs of finite length of vector-valued filters. The corresponding scaling functions and wavelet functions ar...

Let K be an algebraic number field and f a complex-valued function on the ideal class group of K . Then, f extends in a natural way to the set of all non-zero ideals of the ring of integers of K and we can consider the Dirichlet series L(s, f ) = ∑a f (a)N(a)−s which converges for (s) > 1. After extending this function to (s) = 1, and in the case that f does not contain the trivial character, w...

The point-valued time series (PTS) is simply about one value in each or period of the data, but when data have two values at time, suitable called interval-valued (ITS). An example ITS daily close and open stock prices. If are typed linguistic for example, “low increase”, “medium increase” “high a fuzzy-valued being well-known as fuzzy (FTS). aim this study to compare PTS FTS models forecasting...

This introductory article aims to provide a roadmap to many of the interrelated papers in this volume and to a portion of the field of multiple Dirichlet series, particularly emerging new ideas. It is both a survey of the recent literature, and an introduction to the combinatorial aspects of Weyl group multiple Dirichlet series, a class of multiple Dirichlet series that are not Euler products, ...