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.

A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic functions on Green balls is proved. This enables us to generalize many other theorems, on the behaviour of mean values of superharmonic functions over Green spheres, on the Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and on differences of positive superharmonic functions.

Motivated by Candes and Donoho′s work (Candés, E J, Donoho, D L, Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Stat. 30, 784-842 (2002)), this paper is devoted to giving a lower bound of minimax mean square errors for Riesz fractional integration transforms and Bessel transforms.

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 (...

In this letter, the two wavelet families, biorthogonal and Riesz bases are introduced. Biorthogonality for two possible decompositions in these bases, The Riesz stability implies that there exist such that Biorthogonal wavelet bases are related to multiresolution approximations. The direct consequence of the above derivation is tradeoff made between the support size of a wavelet and its number ...

In [B. Han and Z. Shen, SIAM J. Math. Anal., 38 (2006), 530–556], a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in [B. Han and Z. Shen, J. Fourier Anal. Appl., 11 (2005), 615–637]. Motivated by these two papers, we develop in this article a general theory and a construction meth...

We study infinite-dimensional well-posed linear systems with output feedback such that the closed-loop system is well-posed. The generator A of the open-loop system is assumed to be diagonal, i.e., the state space X (a Hilbert space) has a Riesz basis consisting of eigenvectors of A. We investigate when the closed-loop generator A is Riesz spectral, i.e, its generalized eigenvectors form a Ries...

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 .