The results which I will describe arise from joint work with Bill Moran, ru.rd with Dani Berend, Charles Pearce and Andy Pollington. The common theme is the formalism of Riesz product measures, for these have proved to be exceptionally useful in the study of normal numbers. Elsewhere in these proceedings, Tony Dooley describes some of our joint work on the related class of G-measures (G for Gib...

Let G be a countable group of unitary operators on a complex separable Hilbert space H. We give a characterization of biorthogonality among Riesz multiwavelets in terms of certain invariant properties of their associated core spaces. A large family of non-biorthogonal Riesz multiwavelets is exhibited. We also discuss some results on linear perturbation of orthonormal multiwavelets.

In this paper we show that every g-frame for a Hilbert space H can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. We also show that every g-frame can be written as a sum of two tight g-frames with g-frame bounds one or a sum of a g-orthonormal basis and a g-Riesz basis for H . We further give necessary and sufficient conditions on g-Besse...

In this paper, we first discuss about canonical dual of g<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font...

Over the last decades, one field of intense research activity has been the study of extremals of integral functionals. The Riesz-type kind has attracted growing attention and played a crucial role in the resolution of Choquard’s conjecture in a breakthrough paper by E. H. Lieb [1]. The determination of cases of equality in the Riesz-rearrangement inequality has also received a large amount of i...

this paper deals with continuous frames and continuous riesz bases. we introduce continuous riesz bases and give some equivalent conditions for a continuous frame to be a continuous riesz basis. it is certainly possible for a continuous frame to have only one dual. such a continuous frame is called a riesz-type frame [13]. we show that a continuous frame is riesz-type if and only if it is a con...

Let G be a countably infinite group of unitary operators on a complex separable Hilbert space H. Let X = {x1, ..., xr} and Y = {y1, ..., ys} be finite subsets of H, r < s, V0 = spanG(X), V1 = spanG(Y ) and V0 ⊂ V1. We prove the following result: Let W0 be a closed linear subspace of V1 such that V0 ⊕ W0 = V1 (i.e., V0 + W0 = V1 and V0 ∩ W0 = {0}). Suppose that G(X) and G(Y ) are Riesz bases for...

Let (M, g) be a Riemannian manifold. The Riesz transform problem, namely giving conditions on p and on the manifold such that the operator d∆−1/2 – the so-called Riesz transform – is bounded on L, has recently undergone certain progresses. A pioneering result which goes back to 1985 is a theorem of D. Bakry [2] which asserts that if the Ricci curvature of M is non-negative, then the Riesz trans...

More generally, in the first claim we can replace the Lp(Q) norm with any norm || · || on F with the Riesz property, which is that |f | ≤ |g| (pointwise) implies ||f || ≤ ||g||. For the second claim, we can use any norm that has the Riesz property and for which the norm of the constant function f ≡ 1 is 1, which is clearly satisfied by Lp(Q) when Q is a probability measure. With these more gene...