The paper deals with the context of the inner DaniellStone and Riesz representation theorems, which arose within the new development in measure and integration in the book 1997 and subsequent work of the author. The theorems extend the traditional ones, in case of the Riesz theorem to arbitrary Hausdorff topological spaces. The extension enforces that the assertions attain different forms. The ...

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y| with d−2 ≤ s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S i...

Let {X1, . . . ,X2n,T} be a real basis for the Heisenberg Lie algebra hn. In this paper, we calculate the explicit kernels of the Riesz transforms Rj = XjL 1 2 , j = 1, 2, · · · , 2n on the Heisenberg group Hn. Here L = − ∑2n j=1 X 2 j is the sub-Laplacian operator. We show that ∫∞ −∞Rjdt = Rj for j = 1, 2, · · · , 2n where Rj ’s are the regular Riesz transforms on R . We also construct the “mi...

We investigate the relation between the Riesz and the Baez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function R(x) appearing in the Riesz criterion and the sequence c k appearing in the Baez-Duarte formulation. It is shown that R(x) can expressed by c k and vice versa the sequence c k can be obtained from the values of R(x) at integer argumen...

This paper gives an abstract sufficient condition on Riesz basis with parentheses property for the generators of C0-groups in Hilbert spaces whose eigenvalues are comprised of some finite unification of separable sets after taking the algebraic multiplicities into account. The condition is then applied to the closed-loop system of a serially connected string system under joint damping feedbacks...