Riesz potentials and Laplacian of fractal measures in metric spaces are introduced. They deene self{adjoint operators in the Hilbert space L 2 () and the former are shown to be compact. In the euclidean case the corresponding spectral asymptotics are derived by Besov space methods. The inverses of the Riesz potentials are fractal pseudo-diierential operators. For the Laplace operator the spectr...

In this paper, we introduce the diagonal of the modified Riesz distribution defined in 2 , IR  r r . We propose the estimators of different parameters of this new distribution by two method, firstly by the moment method and secondly by the maximum likelihood method. The performance of the both estimators are studied by calculating the Mean Square Error (MSE). KeywordsDiagonal of the modified R...

When does a sequence of spherical codes with “good” spherical cap discrepancy, and “good” separation also have “good” Riesz s-energy? For d > 2 and the Riesz s-energy for 0 < s < d, we consider asymptotically equidistributed sequences of S codes with an upper bound δ on discrepancy and a lower bound ∆ on separation. For such sequences, the difference between the normalized Riesz s-energy and th...

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in R and R. We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classic box-spline direction matrices. These n...

We study linear combinations of exponentials en, λn ∈ Λ in the case where the distance between some points λn tends to zero. We suppose that the sequence Λ is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L(0, T ). Here we prove tha...

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (co...

In this paper we present topology aspects in non-localization results. A well-known such no-go result is the BalianLow theorem that states the generator of a Weyl-Heisenberg Riesz basis cannot be well time-frequency localized. More general, the statement applies to multi WH Riesz bases, or super frames as well. These results turn out to be connected to non-triviality of a complex vector bundle....

Let X and Y be compact Hausdorff spaces, and E, F be Banach lattices. Let C(X,E) denote the Banach lattice of all continuous E-valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism Φ : C(X,E) → C(Y, F ) such that Φf is non-vanishing on Y if and only if f is non-vanishing on X, then X is homeomorphic to Y , and E is Riesz i...

We develop a texture analysis framework to assist radiologists in interpreting high-resolution computed tomography (HRCT) images of the lungs of patients affected with interstitial lung diseases (ILD). Novel texture descriptors based on the Riesz transform are proposed to analyze lung texture without any assumption on prevailing scales and orientations. A global classification accuracy of 78.3%...

We develope a local theory for frames on finite dimensional Hilbert spaces. We show that for every frame (fi) m i=1 for an n-dimensional Hilbert space, and for every ǫ > 0, there is a subset I ⊂ {1, 2, . . . ,m} with |I| ≥ (1 − ǫ)n so that (fi)i∈I is a Riesz basis for its span with Riesz basis constant a function of ǫ, the frame bounds, and (‖fi‖) m i=1 , but independent of m and n. We also con...