Non-spectral self-affine measure problem on the plane domain

The self-affine measure μM,D corresponding to an expanding integer matrix

     

Some remarks on absolute continuity and quantization of probability measures

We introduce a notion of monotonicity of dimensions of measures. We show that the upper and lower quantization dimensions are not monotone. We give sufficient conditions in terms of so-called vanishing rates such that νμ implies . As an application, we determine the quantization dimension of a class of measures which are absolutely continuous w.r.t. some self-similar measure, with the corresponding Radon–Nikodym derivative bounded or unbounded. We study the set of quantization dimensions of measures which are absolutely continuous w.r.t. a given probability measure μ. We prove that the infimum on this set coincides with the lower packing dimension of μ. Furthermore, this infimum can be attained provided that the upper and lower packing dimensions of μ are equal.     

Singularity of certain self-affine measures

The self-affine measure associated with an iterated function system and a weight is uniquely determined. The problem of determining whether a self-affine measure is absolutely continuous or singular has been studied extensively in recent years. In the present paper we consider the singularity of certain self-affine measures. We obtain a sufficient condition for such measures being singular. Two applications of this result are given, which extend several known results in a simple manner.     

Non-spectral problem for a class of planar self-affine measures

The self-affine measure μM,D corresponding to an expanding matrix M∈Mn(R) and a finite subset D⊂Rn is supported on the attractor (or invariant set) of the iterated function system {?d(x)=M⁻¹(x+d)}_d∈D. The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L²(μM,D) and to find them. In the present paper we show that if a,b,c∈Z, |a|>1, |c|>1 and ac∈Z?(3Z),

     

Spectra of a class of self-affine measures

In the present paper we will study the spectral property of a class of self-affine measures under the condition of compatible pair. We first answer a question of Dutkay and Jorgensen concerning the relationship between spectral self-affine measure and compatible pair. We then consider the spectra of Bernoulli convolutions and obtain a sharp result which extends the corresponding result of Jorgensen, Kornelson and Shuman. Finally, we provide a structural property for the integer spectrum of a spectral self-affine measure.     

Non-spectrality of planar self-affine measures with three-elements digit set

The self-affine measure μM,D associated with an affine iterated function system {?d(x)=M−1(x+d)}_d∈D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μM,D have been received much attention in recent years. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L²(μM,D) and to find them. In the present paper we show that for an expanding integer matrix M∈M₂(Z) and the three-elements digit set D given by

     

A class of spectral self-affine measures with four-element digit sets

The self-affine measure _μM,D${\mu }_{M,D}$ associated with an expanding matrix M∈_Mn(Z)$M\in M_n(\mathbb{Z})$ and a finite digit set D⊂Zⁿ$D\subset {\mathbb{Z}}^n$ is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures _μM,D${\mu }_{M,D}$ with four-element digit sets in the higher dimensions (n≥3$n\ge 3$) such that the Hilbert space L²(_μM,D)$L^2({\mu }_{M,D})$ possesses an orthogonal exponential basis. That is, _μM,D${\mu }_{M,D}$ is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.     

Absolute continuity of Wasserstein geodesics in the Heisenberg group

In this paper we answer to a question raised by Ambrosio and Rigot [L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261-301] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the end-points is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolute continuity of Wasserstein geodesic also holds for Alexandrov spaces with curvature bounded from below.     

Absolute continuity of a vector-valued measure

The necessary and sufficient condition is proved for absolute continuity of a vector-valued measure in the sense of Bartle.     

Local dimensions of measures on infinitely generated self-affine sets

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space.     

Absolute continuity of the distribution of some Markov geometric series

Let (∈n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift space ∑A defined by Fibonacci incident matrix A.We consider the measure μλ of the probability distribution of the random series ∑∞n=0 εnλn (0 ＜λ＜ 1).It is proved that μλ is singular if λ∈ (0,√5-1/2) and that μλ is absolutely continuous for almost all λ∈ (√5-1/2,0.739).     

On the Hencl's notion of absolute continuity

We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in , J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].     

Substitution-based structures with absolutely continuous spectrum

By generalising Rudin’s construction of an aperiodic sequence, we derive new substitution-based structures which have a purely absolutely continuous diffraction measure and a mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.     

On the absolute continuity of a class of invariant measures

Let be a compact connected subset of , let , be contractive self-conformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigen-equation

for some 0$">. We prove that if the attractor is an -set and is absolutely continuous with respect to , the Hausdorff -dimensional measure restricted on the attractor , then is absolutely continuous with respect to (i.e., they are equivalent). A special case of the result was considered by Mauldin and Simon (1998). In another direction, we also consider the -property of the Radon-Nikodym derivative of and give a condition for which is unbounded.

     

Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of C(X,R^k), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,R^k).     

Absolute Continuity and the Uniqueness of the Constructive Functional Calculus

The constructive functional calculus for a sequence of commuting selfadjoint operators on a separable Hilbert space is shown to be independent of the orthonormal basis used in its construction. The proof requires a constructive criterion for the absolute continuity of two positive measures in terms of test functions. Mathematics Subject Classification: 03F60, 46S30, 47S30.     

Absolute continuity of multivariate distributions of class L

It is shown that every genuinely d-dimensional distribution of class L on R^d is absolutely continuous. This extends the known fact in one dimension to all finite dimensions.     

Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μM,D corresponding to the expanding integer matrix

     

Absolute continuity of operator-self-decomposable distributions on Rd

It is shown that every genuinely d-dimensional operator-self-decomposable distribution is absolutely continuous.