Recently, we presented a new image pyramid, called the Riesz pyramid, that uses the Riesz transform to manipulate the phase in non-oriented sub-bands of an image sequence to produce real-time motion-magnified videos. In this report we give a quaternionic formulation of the Riesz pyramid, and show how several seemingly heuristic choices in how to use the Riesz transform for phase-based video mag...

In this paper, we focus on the structured multi-frame vectors in Hilbert $C^*$-modules. More precisely, it will be shown that the set of all complete multi-frame vectors for a unitary system can be parameterized by the set of all surjective operators, in the local commutant. Similar results hold for the set of all complete wandering vectors and complete multi-Riesz vectors, when the surjective ...

Compactly supported Riesz wavelets are of interest in several applications such as image processing, computer graphics and numerical algorithms. In this paper, we shall investigate compactly supported MRA Riesz multiwavelet bases in L2(R). An algorithm is presented to derive Riesz multiwavelet bases from refinable function vectors. To illustrate our algorithm and results in this paper, we prese...

The generalized Bochner-Riesz operator SR,λ may be defined as Sf(x) = F−1 [( 1− ρ R )λ + f̂ ] (x) where ρ is an appropriate distance function and F−1 is the inverse Fourier transform. The sharp bound ‖Sf‖L4(R2×R2) ≤ C‖f‖L4(R2×R2) is shown for the distance function ρ(ξ′, ξ”) = max{|ξ′|, |ξ′′|}. This is a rough distance function corresponding to the R4 cylinder analog {(x1, x2, x3, x4) ∈ R4, x1+ x...

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...

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.