Let L be an Archimedean Riesz space with a weak order unit u. A sufficient condition under which Dedekind [σ-]completeness of the principal ideal Au can be lifted to L is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of C(X)-spaces. Similar results are obtained for the Riesz spaces Bn(T ), n = 1, 2, . . . , of all functions o...

We consider a functional on the Wiener space which is smooth and not degenerated in Malliavin sense and we give a criterion for the strict positivity of the density, that we can use to state lower bounds as well. The results are based on the representation of the density in terms of the Riesz transform introduced in Malliavin and Thalmaier bib:[M.T] [16] and on the estimates of the Riesz transf...

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and of Hardy and Littlewood, based on the order of certain entire functions on the positive real axis, are here embedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved ...

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...

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...

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%...