In the paper the discrete version of the Morse’s singularity condition is established. This condition ensures that the discrete functional over the unbounded interval is positive semidefinite on the class of the admissible functions. Two types of admissibility are considered.

We prove sharp Lp(w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the Ap characteristic of w for all 1 < p <∞. This implies the same sharp inequalities for the classical Lusin area integral S(f ), the Littlewood–Paley g-function, and their continuous analogs Sψ and gψ . Also, as a corollary, we obtain sharp weighted inequalities for any conv...

Let A : D(A) ⊆ H → H be an injective self-adjoint operator and let τ : D(A) → X, X a Banach space, be a surjective linear map such that ‖τφ‖X ≤ c ‖Aφ‖H. Supposing that Range (τ ) ∩ H = {0}, we define a family AτΘ of self-adjoint operators which are extensions of the symmetric operator A|{τ=0} . Any φ in the operator domain D(A τ Θ) is characterized by a sort of boundary conditions on its univoc...

In this paper, switched controllers are designed for a class of nonlinear discrete singular systems and a class of discrete singular bilinear systems. An invariant principle is presented for such switched nonlinear singular systems. The invariant principle and the switched controllers are used to globally stabilize a class of singular bilinear systems that are not open-loop stable.

This paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our analysis of dispersive errors indicates that the DSC algorithm yields a dispersion vanishing scheme. ...

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective cor...

This paper is concerned with the efficient implementation of transparent boundary conditions (TBCs) for wide angle parabolic equations (WAPEs) assuming cylindrical symmetry. In [1] a discrete TBC of convolution type was derived from the fully discretized whole–space problem that is reflection–free and yields an unconditionally stable scheme. Since the discrete TBC includes a convolution with re...

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective corres...

For j = 1, ..., n, letΩj be open sets of the complex plane and let φj be holomorphic functions on Ωj such that φ′′ j does not vanish identically on Ωj . We consider φ (z1, ..., zn) = φ1 (z1) + ...+ φn (zn). We characterize the pairs (p, q) such that the convolution operator with the surface measure supported on a compact subset of the graph of φ is p− q bounded.

We study the middle convolution of local systems in the setting of singular and étale cohomology. We give a motivic interpretation of the middle convolution in the étale case and prove an independence-of-`-result which yields a description of the determinant. We employ these methods to realize special linear groups regularly as Galois groups over Q(t). In an appendix to this article, written jo...