Abstract: We show that a non-negative Hamiltonian operator whose domain contains a maximal uniformly positive subspace is bounded.

Abstract: We consider equidistant discrete splines S(j), j ∈ Z, which may grow as O(|j|s) as |j| → ∞. Such splines present a relevant tool for digital signal processing. The Zak transforms of Bsplines yield the integral representation of discrete splines. We define the wavelet space as a weak orthogonal complement of the coarse-grid space in the fine-grid space. We establish the integral representation of the elements of the wavelet space. We define and characterize the wavelets whose shifts form bases of the wavelet space. By this means we design a wide library of bases for the space of discrete-time signals of power growth construct multiscale representation of this space. We provide formulas for processing such the signals by discrete spline wavelets. Constructed bases are at the same time the Riesz bases for the space l2.

Abstract: The space L 2(G;ℂ m ) of Clifford-algebra-valued functions in bounded domains G of ℝ m is decomposed into the orthogonal sum of the subspace of poly-left-monogenic functions of arbitrary order k≥1 and its orthogonal complement and as well into the orthogonal sum of the subspace of polyharmonic functions of arbitrary order k≥1 and its orthogonal complement. The complementary subspaces are given explicitly. In the particular case m=2, complex functions are involved. Although this case has to be treated separately, the results are as before. The proofs are based on proper higher-order Cauchy–Pompeiu formulas and Green functions for powers of the Laplacian.

Abstract: In this article we investigate spaces of functions defined in a domain Ω ⊂ R with values in the Clifford algebra R n. According to an inner product an orthogonal decomposition is proved. By this decomposition, we obtain a subspace A 2(Ω) of regular functions with respect to the Dirac operator. In the orthogonal complement the Dirac equation with homogeneous boundary values is solvable. The decomposition can be proved in two ways: by a reflection principle and by Sobolev's regularity theorem. It will turn out, that the existence of the orthogonal decomposition and Sobolev's theorem is equivalent. So also a reflection principle will be proved, which describes the jump behavior of a Cauchy type integral. By the reflection principle, a countable dense subset of A 2(Ω) can be obtained. Further considerations lead to a minimal generating system, by which the Bergman kernel function can be obtained. As a conclusion we also obtain Runge's theorem.

Abstract: We present explicit outer boundary conditions for the canonical variables of general relativity. The conditions are associated with the causal evolution of a finite Cauchy domain, a so-called quasilocal boost, and they suggest a consistent scheme for modelling such an evolution numerically. The scheme involves a continuous boost in the spacetime orthogonal complement ⊥Tp(B) of the tangent space Tp(B) belonging to each point p on the system boundary B. We show how the boost rate may be computed numerically via equations similar to those appearing in canonical investigations of black-hole thermodynamics (although here holding at an outer two-surface rather than the bifurcate two-surface of a Killing horizon). We demonstrate the numerical scheme on a model example, the quasilocal boost of a spherical three-ball in Minkowski spacetime. Developing our general formalism with recent hyperbolic formulations of the Einstein equations in mind, we use Anderson and York's 'Einstein–Christoffel' hyperbolic system as the evolution equations for our numerical simulation of the model.

Abstract: In this short paper we study the problem of existence of a controlled behavior that is strictly dissipative with respect to a quadratic supply rate. The relation between strictness and the rank of a suitable qudratic differential form that couples the dissipativity properties of the hidden behavior and the orthogonal complement of the plant behavior is analyzed.

Abstract: Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.