In this paper we construct C continuous piecewise quadratic hierarchical bases on Powell–Sabin triangulations of arbitrary polygonal domains in R. Our bases are of Lagrange type instead of the usual Hermite type and we prove that they form strongly stable Riesz bases for the Sobolev spaces Hs(Ω) with s ∈ (1, 5 2 ). Especially the case s = 2 is of interest, because we can use the corresponding h...

Let A be a compact set in Rp of Hausdorff dimension d. For s ∈ (0, d), the Riesz s-equilibrium measure μs is the unique Borel probability measure with support in A that minimizes Is(μ) := " 1 |x − y|s dμ(y)dμ(x) over all such probability measures. If A is strongly (Hd , d)-rectifiable, then μs converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s...

Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler– Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability...

Estimation of L norms of Fourier multipliers is known to be hard. It is usually connected to some interesting types of PDE, see several such PDE for several Fourier multipliers on the line in a recent paper of Kalton and Verbitsky [13]. Sometimes, but much more rarely, one can establish sharp L estimates for Fourier multipliers in several variables. Riesz transforms are examples of success. The...

We introduce multipulse multicarrier (MPMC) modulation, a wireless communication scheme that augments traditional singlepulse multicarrier systems by using multiple pulses at the transmitter and the receiver. The mathematical foundation of MPMC systems is established by the novel concept of multipulse Gabor Riesz bases. We adapt Zak-Fourier domain tools previously developed for multiwindow Gabo...

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...

We study the Riesz $(a,p)$-capacity of so called Dobi\'nski set. characterize values parameters $a$ and $p$ for which $(a,p)$-Riesz capacity set is positive. In particular we show that has positive logarithmic capacity, thus answering a question Dayan, Fernand\'ez Gonz\'alez. approach problem by considering dyadic analogues $(a,p)$-capacities seem to be better adapted problem.

A model is considered for turbulent diffusion which consists of a Riesz space fractional derivative to describe the turbulent phenomenon and also includes advection and classical diffusion. We present a first order explicit numerical method and a second order implicit numerical method to solve our problem and prove convergence results for both methods, including the derivation of stability cons...

We consider the natural generating system for a cyclic subspace of a Hilbert space generated by a dual integrable unitary representation of a countable abelian group. We prove, under mild hypothesis, that whenever the generating system is a quasi-greedy basis it must also be an unconditional Riesz basis. A number of applications to Gabor systems and to general Vilenkin systems are considered. I...