Riesz Products Research Articles

Mixing and asymptotic distribution modulo 1

AbstractIf μ is a probability measure which is invariant and ergodic with respect to the transformationx↦qxon the circle ℝ/ℤ, then according to the ergodic theorem, {qnx} has the asymptotic distribution μ for μ-a.e.x. On the other hand, Weyl showed that when μ is Lebesgue measure, λ, and {mj} is an arbitrary sequence of integers increasing strictly to ∞, the asymptotic distribution of {mjx} is λ for λ-a.e.x. Here, we investigate the asymptotic distributions of {mjx} μ-a.e. for fairly arbitrary {mj} under some strong mixing conditions on μ. The result is a kind of stable ergodicity: the distributions are obtained from simple operations applied to μ. The ideas extend to the situation of a sequence of transformationsx↦qnxwhere invariance is not present. This gives us information about many Riesz products and Bernoulli convolutions. Finally, we apply the theory to resolve some questions aboutH-sets.

On Riesz product measures; mutual absolute continuity and singularity

On Riesz product measures; mutual absolute continuity and singularity

Riesz products, Hausdorff dimension and normal numbers

We say a real number x is normal to base r if the sequence is uniformly distributed modulo [0, 1]. Pollington[10] has proved the following result concerning the normality of numbers.

Most Riesz Product Measures are L p -Improving

Most Riesz Product Measures are L p -Improving

Riesz products and spectrum of Mackey actions

Riesz products and spectrum of Mackey actions

Most Riesz product measures are 𝐿^{𝑝}-improving

A Borel measure μ \mu on a compact abelian group G G is L p {L^p} -improving if, given p > 1 p > 1 , there is a q = q ( p , μ ) > p q = q(p,\mu ) > p and a K = K ( p , q , μ ) > 0 {\text {a}}\;K = K(p,q,\mu ) > 0 such that ‖ μ ∗ f ‖ q ≤ K ‖ f ‖ p {\left \| {\mu * f} \right \|_q} \leq K{\left \| f \right \|_p} for each f f in L p ( G ) {L^p}(G) . Here the L p {L^p} -improving Riesz product measures on infinite compact abelian groups are characterized by means of their Fourier transforms.

Riesz Products and Normal Numbers

Riesz Products and Normal Numbers

Sidon sets and Riesz products

Let G be a compact abelian group and Γ the dual group. It is shown that if Δ⊂Γ is a Sidon set, then the interpolating measures on Λ can be obtained as mean of Riesz products. If Λ is a Sidon set tending to infinity, Λ is of first type. Our approach yields in fact elementary proofs of certain characterizations of Sidonicity obtained in G. Pisier, C.R.A.S., Paris Ser. A, 286 (1978), 1003–1006 – Math. Anal. and Appl., Part B, Advances in Math., Suppl. Sts. vol. 7, 685-726 – preprint, using random Fourier series.

Riesz Products are Basic Measures

Riesz Products are Basic Measures

Singularity of generalised Riesz products

AbstractCriteria for singularity and absolute continuity analogous to Zygmund's for Riesz products, are demonstrated for those generalized products introduced by Modica and Mortola.

Singular Measures and Riesz Products

Singular Measures and Riesz Products

The interior of the Šilov boundary of M( G) is trivial

Let G be a non-discrete locally compact abelian group, and let M( G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the Šilov boundary of M( G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.

Spectra of singular measures as multipliers on Lp

Stein's theorem on the interpolation of a family of operators between two analytic spaces is generalized, both to a multiply connected domain and to an interpolation between more than two spaces. The theorem is then applied to get setwise upper bounds for spectra of convolution operators on L p of the circle. In particular the spectra of operators given by convolution by Cantor-Lebesgue-type measures on L p are determined. The same is done for certain Riesz products. These results are used to derive a result on translation-invariant subspaces of L p of the circle.

Sur la singularité des produits de Riesz et des mesures spectrales associées à la somme des chiffres

Using a property of generalized characters, we first prove that two Riesz products with constant coefficients and distinct Fourier spectra are mutually singular. IfSr(n) denotes the sum of digits in ther-adic representation of the integern, the same technique allows us to establish the mutual singularity of the spectral measures of the sequences: α(n)=exp[2iπaSp(n)],β(n)=exp[2iπbSq(n)], for every pair of integersp≠q, a, b being real numbers such thata(p−1)∉ {tiZ} andb(q−1)∉Z. This result has been proved by T. Kamae whenp andq are two relatively prime integers.

On dichotomy of Riesz products

Background. Riesz products are very useful for the construction of singular measures on compact, Abelian groups. Under some circumstances, two Riesz products are either equivalent or singular in the measure-theoretic sense. Exact knowledge of these circumstances has been of major interest ever since the 1930s, when Riesz's famous example (8) was recognized as a fertile source of examples of singular continuous measures. Zygmund(11) showed that any Riesz product over a Hadamard dissociate subset of ℕ is either a square integrable function or singular with respect to Lebesgue measure. Hewitt–Zuckerman(4) generalized these products to all compact, Abelian groups, introducing the notion of a dissociate subset. They extended Zygmund's result in certain cases. The next major step was taken by Brown–Moran(3) and Peyrière(6), (7), who showed that two Riesz productsare mutually singular ifThe author (9) has improved another result of Brown–Moran (3) by showing that µa and µb are equivalent if

Functional Calculus and Positive-Definite Functions

For a LCA group G with dual group Ĝ, let $D(G) = D(\hat G)$ denote the convex (not closed) hull of $\{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\}$. The set $D(G)$ is the natural domain for functions that operate by composition from the class, $P{D_1}(\hat G)$, of Fourier-Stieltjes transforms of probability measures on G to $B(\hat G)$, the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of $D(G)$. In §1, we show (1) if F operators from $P{D_1}(G)$ to $B(G)$ and G is compact, then $K(z) = {\lim _{t \to {1^ - }}}F(tz)$ exists for all $z \in D(G)$ and K operates from $P{D_1}(\hat G)$ to $B(\hat G)$; (2) if F operates from $P{D_1}(\hat G)$ to $PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G)$ and G is compact, then K operates from $P{D_1}(\hat G)$ to $PD(\hat G)$, and so also does $F - K$; (3) if $G = {{\mathbf {D}}_q},q \geqslant 2$, and F operates from $P{D_1}(\hat G)$ to $B(\hat G)$, then $F = K$ on $D(G) \cap \{ z:|z| < 1\}$. This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from $P{D_1}(\hat G)$ to $B(\hat G)$. In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.

Functional calculus and positive-definite functions

For a LCA group G with dual group Ĝ, let D ( G ) = D ( G ^ ) D(G) = D(\hat G) denote the convex (not closed) hull of { ⟨ x , γ ⟩ : x ∈ G , γ ∈ G ^ } \{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\} . The set D ( G ) D(G) is the natural domain for functions that operate by composition from the class, P D 1 ( G ^ ) P{D_1}(\hat G) , of Fourier-Stieltjes transforms of probability measures on G to B ( G ^ ) B(\hat G) , the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of D ( G ) D(G) . In §1, we show (1) if F operators from P D 1 ( G ) P{D_1}(G) to B ( G ) B(G) and G is compact, then K ( z ) = lim t → 1 − F ( t z ) K(z) = {\lim _{t \to {1^ - }}}F(tz) exists for all z ∈ D ( G ) z \in D(G) and K operates from P D 1 ( G ^ ) P{D_1}(\hat G) to B ( G ^ ) B(\hat G) ; (2) if F operates from P D 1 ( G ^ ) P{D_1}(\hat G) to P D ( G ^ ) = ∪ r &gt; 0 r P D 1 ( G ^ ) PD(\hat G) = { \cup _{r &gt; 0}}rP{D_1}(\hat G) and G is compact, then K operates from P D 1 ( G ^ ) P{D_1}(\hat G) to P D ( G ^ ) PD(\hat G) , and so also does F − K F - K ; (3) if G = D q , q ⩾ 2 G = {{\mathbf {D}}_q},q \geqslant 2 , and F operates from P D 1 ( G ^ ) P{D_1}(\hat G) to B ( G ^ ) B(\hat G) , then F = K F = K on D ( G ) ∩ { z : | z | &gt; 1 } D(G) \cap \{ z:|z| &gt; 1\} . This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from P D 1 ( G ^ ) P{D_1}(\hat G) to B ( G ^ ) B(\hat G) . In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.

Riesz Products and Generalized Characters

Riesz Products and Generalized Characters

On orthogonality of Riesz products

A typical Riesz product on the circle is the weak* limitwhere – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

Riesz products on non-commutative groups

Riesz products on non-commutative groups