Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that $E={\mathbf{R}}^{m+1}$ is given the presheaf of solutions of$$\sum _{i=1}^m\frac{\partial u}{\partial x_i}=\frac{\partial u}{\partial x_{m+1}}$$and $B$ is a subset of ${\mathbf{R}}^m\times [-\infty ,0]$ satisfying$$\left\{\right(\alpha x,{\alpha }^{2}t):(x,t)\in B,x\in {\mathbf{R}}^m,t\in \mathbf{R}\}\subset B$$for $\alpha \>0$ arbitrarily small. Then $B$ is thin at 0 if and only if $B$ is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several approximation theorems on reducing and balayage...

We give a complete characterization of the positive trigonometric polynomials $Q(\theta ,\varphi )$ on the bi-circle, which can be factored as $Q(\theta ,\varphi )=|p(e^{i\theta },e^{i\varphi }){|}^2$ where $p(z,w)$ is a polynomial nonzero for $\left|z\right|=1$ and $\left|w\right|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4{\pi }^{2}Q(\theta ,\varphi )}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...

This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Wentzell boundary conditions in the characteristic case. Our results may be stated as follows: We can construct Feller semigroups corresponding to a diffusion phenomenon including absorption, reflection, viscosity, diffusion along the boundary and jump at each point of the boundary.

A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak$\check{\mathrm{s}}$ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus and complex, real and harmonic analysis. It also leads to several results of independent...

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and $L^1$ contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in $L^1$ and $L^{\infty }$, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive $L^1$ convergence without any convergence rate....