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.

Let λ be a positive number, and let (xj : j ∈ Z) ⊂ R be a fixed Riesz-basis sequence, namely, (xj) is strictly increasing, and the set of functions {R ∋ t 7→ ej : j ∈ Z} is a Riesz basis (i.e., unconditional basis) for L2[−π,π]. Given a function f ∈ L2(R) whose Fourier transform is zero almost everywhere outside the interval [−π, π], there is a unique square-summable sequence (aj : j ∈ Z), depe...

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (...

The only quadrature operator of order two on L2(R) which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In thi...

The Riesz transform is a natural multidimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of...

Riesz distributions are relatively invariant distributions supported by the closure of a homogeneous cone . In this paper, we clarify the positivity condition of Riesz distributions by relating it to the orbit structure of . Moreover each of the positive Riesz distributions is described explicitly as a measure on an orbit in .

The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform o...

In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets form Riesz bases of certa...

This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler–Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrumdetermined growth condition,...