with n0 ≡ N. The finite harmonic sums form classes which are given by their transcendentality t = ∑m l=1 |kl|. This term was chosen to characterize the behavior of these sums for N → ∞. In this case the values of the sums, if existing, are described by a transcendental number of rank t, i.e. log 2, ζ(2), ζ(3) or log 2·ζ(2), etc. for t = 1, 2, 3, ... All harmonic sums have a linear representatio...

It is known that there is no nonconstant bounded harmonic map from the Euclidean space R to the hyperbolic space H. This is a particular case of a result of S.-Y. Cheng. However, there are many polynomial growth harmonic maps from R to H by the results of Z. Han, L.-F. Tam, A. Treibergs and T. Wan. One of the purposes of this paper is to construct harmonic maps from R to H by prescribing bounda...

In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...

In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...

We revisit the radial oscillator from standpoint of a free realization. By using oscillator, namely, creation/annihilation operators harmonic we construct an operator that maps eigenfunctions to those oscillator. As polynomial part this relation, obtain Hermite polynomials Laguerre polynomials.

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in Rn such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in Rn can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fi...

For any manifold with polynomial volume growth, we show that the dimension of space ancient caloric functions growth is bounded by degree times harmonic same growth. As a consequence, get sharp bound for on where Yau’s 1974 conjecture about holds.