We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications t...

In this article we obtain families of frames for the space Bω of functions with band in [−ø, ø] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of A. Ron and Z. Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates o...

We prove that the Atiyah-Singer Dirac operator / Dg in L 2 depends Riesz continuously on L∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g → / Dg(1 + / D 2 g) − 12 depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón’s first commu...

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also symmetric, there exists a Riesz basis exponential functions in the space $L^2(\Omega)$. The result new $d$ greater than one.

Assume that G is a nilpotent Lie group. Denote by [Formula: see text] the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class [Formula: see text] for some [Formula: see text] and D is the dimension at infinity of G. Let [Formula: see text] be the Riesz transform associated with L. In this paper we obtain some estimates for the...

Abstract We formulate a generalization of Riesz-type criteria in the setting L -functions belonging to Selberg class. obtain criterion which is sufficient for grand Riemann hypothesis (GRH) satisfying axioms class without imposing Ramanujan on their coefficients. also construct subclass and prove necessary GRH this subclass. Identities Ramanujan–Hardy–Littlewood type are established setting, sp...

Let (M, g) be a complete Riemannian manifold which satisfies a Sobolev inequality of dimension n, and on which the volume growth is comparable to the one of R for big balls; if there is no non-zero L harmonic 1-form, and the Ricci tensor is in L n 2 −ε ∩ L∞ for an ε > 0, then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian acting on 1-forms. This allows us to prove that, ...

We prove L estimates for the maximal Riesz transform in terms of the Riesz transform itself, for 1 < p ≤ ∞. We show that the corresponding weak L1 estimate fails for the maximal Riesz transform, but surprisingly does hold for the maximal Beurling transform.

Suucient conditions for the convergence in distribution of an inn-nite convolution product 1 2 : : : of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy T = where T is a reenement operator constructed from a measure and a dilation automorphism A. The existence of A implies G is nilpoten...