Normalized Lebesgue Measure Research Articles

Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets

Abstract Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the s-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where s is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.

Uniformly convergent Fourier series with universal power parts on closed subsets of measure zero

Given a closed subset E of Lebesgue measure zero on the unit circle T there is a function f on T with uniformly convergent symmetric Fourier series Sn(f,ζ)=∑k=−nnfˆ(k)ζk⇉Tf(ζ),such that for every continuous function g on E, there is a subsequence of partial power sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. Here fˆ(k)=∫Tζ̄kf(ζ)dm(ζ),and m is the normalized Lebesgue measure on T.

Systems of rank one, explicit Rokhlin towers, and covering numbers

Rotations f_alpha of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle alpha are known to be dynamical systems of rank one. This is equivalent to the property that the covering number F^*(f_alpha ) of the dynamical system is one. In other words, there exists a basis B such that for arbitrarily high h, an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower (f_alpha ^kB)_{k=0}^{h-1}. Although B can be chosen with diameter smaller than any fixed varepsilon > 0, it is not always possible to take an interval for B but this can only be done when the partial quotients of alpha are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when B is the union of n_B in {mathbb {N}} disjoint intervals. This question has been answered in the case n_B =1 by Checkhova, and here we address the general situation. If n_B = 2, we give a precise formula for the maximum proportion. Furthermore, we show that for fixed alpha , the maximum proportion converges to 1 when n_B rightarrow infty . Explicit lower bounds can be given if alpha has constant partial quotients. Our approach is inspired by the construction involved in the proof of the Rokhlin lemma and furthermore makes use of the three gap theorem.

Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

AbstractWe consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

Stability of the mean value formula for harmonic functions in Lebesgue spaces

Let D be an open subset of $${\mathbb {R}}^n$$ with finite measure, and let $$x_0 \in D$$ . We introduce the p-Gauss gap of D w.r.t. $$x_0$$ to measure how far are the averages over D of the harmonic functions $$ u \in L^p(D)$$ from $$u(x_0)$$ . We estimate from below this gap in terms of the ball gap of D w.r.t. $$x_0$$ , i.e., the normalized Lebesgue measure of $$D \setminus B$$ , being B the biggest ball centered at $$x_0$$ contained in D. From these stability estimates of the mean value formula for harmonic functions in $$L^p$$ -spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space $$W^{1,p'}$$ , where $$p'$$ is the conjugate exponent of p.

Isotropic measures and maximizing ellipsoids: Between John and Loewner

We define a one-parametric family of positions of a centrally symmetric convex body K K which interpolates between the John position and the Loewner position: for r > 0 r>0 , we say that K K is in maximal intersection position of radius r r if Vol n ( K ∩ r B 2 n ) ≥ Vol n ( K ∩ r T B 2 n ) \textrm {Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm {Vol}_{n}(K\cap rTB_{2}^{n}) for all T ∈ S L n T\in \rm {SL}_{n} . We show that under mild conditions on K K , each such position induces a corresponding isotropic measure on the sphere, which is simply the normalized Lebesgue measure on r − 1 K ∩ S n − 1 r^{-1}K\cap S^{n-1} . In particular, for r M r_{M} satisfying r M n κ n = Vol n ( K ) r_{M}^{n}\kappa _{n}=\textrm {Vol}_{n}(K) , the maximal intersection position of radius r M r_{M} is an M M -position, so we get an M M -position with an associated isotropic measure. Lastly, we give an interpretation of John’s theorem on contact points as a limit case of the measures induced from the maximal intersection positions.

ON THE ONE-WAY SPECIFICATION PROPERTY AND LARGE DEVIATIONS FOR SYSTEMS WITH NON-DENSE ERGODIC MEASURES

We introduce a weaker form of the specification property, called the ‘one-way specification property', which holds for a broad class of systems whose ergodic measures are not dense in the set of invariant measures, including Smale's horseshoe map, DA map, and (symbolic spaces of) (–β)-transformations for Yrrap β > 1. As an application, we show that these three examples satisfy a level-2 large deviation principle with the normalized Lebesgue measure.

The circular law for random regular digraphs with random edge weights

We consider random [Formula: see text] matrices of the form [Formula: see text], where [Formula: see text] is the adjacency matrix of a uniform random [Formula: see text]-regular directed graph on [Formula: see text] vertices, with [Formula: see text] for some fixed [Formula: see text], and [Formula: see text] is an [Formula: see text] matrix of i.i.d. centered random variables with unit variance and finite [Formula: see text]th moment (here ∘ denotes the matrix Hadamard product). We show that as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges weakly in probability to the normalized Lebesgue measure on the unit disk.

$$L^{p}$$ L p -Norms and Mahler’s Measure of Polynomials on the n-Dimensional Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.

Stochastic stability of measures in gradient systems

Stochastic stability of a compact invariant set of a finite dimensional, dissipative system is studied in our recent work “Concentration and limit behaviors of stationary measures” (Huang et al., 2015) for general white noise perturbations. In particular, it is shown under some Lyapunov conditions that the global attractor of the systems is always stable under general noise perturbations and any strong local attractor in it can be stabilized by a particular family of noise perturbations. Nevertheless, not much is known about the stochastic stability of an invariant measure in such a system. In this paper, we will study the issue of stochastic stability of invariant measures with respect to a finite dimensional, dissipative gradient system with potential function f. As we will show, a special property of such a system is that it is the set of equilibria which is stable under general noise perturbations and the set Sf of global minimal points of f which is stable under additive noise perturbations. For stochastic stability of invariant measures in such a system, we will characterize two cases of f, one corresponding to the case of finite Sf and the other one corresponding to the case when Sf is of positive Lebesgue measure, such that either some combined Dirac measures or the normalized Lebesgue measure on Sf is stable under additive noise perturbations. However, we will show by constructing an example that such measure stability can fail even in the simplest situation, i.e., in 1-dimension there exists a potential function f such that Sf consists of merely two points but no invariant measure of the corresponding gradient system is stable under additive noise perturbations. Crucial roles played by multiplicative and additive noise perturbations to the measure stability of a gradient system will also be discussed. In particular, the nature of instabilities of the normalized Lebesgue measure on Sf under multiplicative noise perturbations will be exhibited by an example.

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

For an interval E=[a,b] on the real line, let μ be either the equilibrium measure, or the normalized Lebesgue measure of E, and let Vμ denote the associated logarithmic potential. In the present paper, we construct a function f which is analytic on E and possesses four branch points of second order outside of E such that the family of the admissible compacta of f has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V−μ.

On the growth of the polynomial entropy integrals for measures in the Szegő class

Let σ be a probability Borel measure on the unit circle T and {ϕn} be the orthonormal polynomials with respect to σ. We say that σ is a Szegő measure, if it has an arbitrary singular part σs, and ∫Tlogσ′dm>−∞, where σ′ is the density of the absolutely continuous part of σ, m being the normalized Lebesgue measure on T. The entropy integrals for ϕn are defined as ϵn=∫T|ϕn|2log|ϕn|dσ. It is not difficult to show that ϵn=o¯(n). In this paper, we construct a measure from the Szegő class for which this estimate is sharp (over a subsequence of n’s).

Erratum to: Closed-Range Composition Operators on $${\mathbb{A}^2}$$ and the Bloch Space

We thank Nina Zorboska for pointing out an error in the proof of [2, Proposition 3.1]; a faux pas in our application of Julia’s Theorem. Indeed, we find that Proposition 3.1 does not hold in general (see Example 2.4 in this erratum). This impacts two operator-theoretic results in Section 3 of [2]; namely, Theorems 3.5 and 3.7. Theorem 3.5 (in [2]) does not hold in general. While statements (i) and (ii) of Theorem 3.7 are equivalent (cf. [1, Theorem 2.4]), statement (iii) implies, but is not a consequence of (i) and (ii) (in general). Both of these theorems hold under an additional hypothesis. We make all of this clear in our work here. And, although [2, Corollary 3.6] was established using Theorem 3.5, we find that this corollary holds in general; see the proof of Corollary 1.6 in this erratum. In order that our presentation be somewhat self-contained, we begin by reviewing the terminology of [2]. Let D denote the unit disk {z : |z| < 1} and let T denote its boundary {z : |z| = 1}. Let A denote normalized two-dimensional Lebesgue measure on D and let m denote normalized Lebesgue measure on T; normalized so that these are probability measures. The Bergman space A is the Hilbert space of functions f that are analytic in D such that

Norm inequality of AP+BQ for selfadjoint projections P and Q with PQ=0

Let A,B,P and Q be bounded operators on a Hilbert space L which P and Q are selfadjoint projections with PQ = 0. We study the norm of AP+BQ using a Hankel type operator : HB∗A = QB ∗ A | PL . Let L be a Hilbert space and let H and K denote closed subspaces of L such that H is orthogonal to K. P and Q denote orthogonal projections with PQ = 0. We suppose H = PL and K = QL . For a bounded linear operator X on L ,w e define a Hankel type operator HX from H to K : HX f = Q( Xf )( f ∈ H). In a very special case, Yamamoto and the author (3) have established a theorem about the norm of AP+BQ .L etm be the normalized Lebesgue measure on the unit circle T.F orp = 2, ∞, L p (T) denotes the usual Lebesgue space on T and H p (T) denotes the usual Hardy space on T.L etL =L 2 (T),H = H 2 (T) and K = L 2 (T) � H 2 (T). Suppose A =Mα and B = M β are multiplication operators on L 2 (T) where α and β are in L ∞ (T). Then the following formula is known (3).

Path Connected Components in the Space of Weighted Composition Operators on the Disk Algebra

Let H2 be the Hardy space and H°° the Banach space of bounded analytic functions on the open unit disk ID with the supremum norm || • Hoc. For / £ H°°, we denote by /* the radial limit of / defined a.e. on the boundary 3D of P. We write {|/*| = 1} = {el° £ 3D : \f*(ete) — 1}. Let m be the normalized Lebesgue measure on 3D. We denote by 5#* the set of analytic self-maps of D and Sh^.o — G Sh=c : Hvlloo = !}• For p £ Shthe composition operator C^ on H2 is defined by C^f = f o tp for / £ H2. Let C(H2) be the space of composition operators on H2 with the operator norm. The most interesting subject in the study of C(H2) is called the connected component problem, describing the connected component containing a given Cv. It was first studied by Berkson [2]. He showed that if p € 0, then is an isolated point in C(H2). Around 1990, MacCluer [13] and Shapiro and Sundberg [19] revisited the connected component problem on H2. Succeedingly, Bourdon [3], Gallardo-Gutierrez, Gonzalez, Nieminen and Saksman [6] and Moorhouse and Toews [16] followed. Still it remains unclear the connected component problem on H2 (see [4; 17; 18]). Similarly we have the space C(H°°) of composition operators on H°° with the

On the $p$-norm of the Berezin transform

In this short note, the norm of Berezin transform, acting on $L^{p}(\mathbb{B}_{n})$, is determined to be \[\bigl\|B_{\mathbb{B} _{n}}:L^{p}(\mathbb{B}_{n})\to L^{p}(\mathbb{B}_{n})\bigr\|=\frac{1}{p}\prod_{k=1}^{n}\biggl(1+\frac{1}{kp}\biggr)\frac{\pi}{\sin(\pi/p)}.\] This extends a result of Dostanić (J. Anal. Math. 104 (2008) 13–23) to several complex variables.

Bounded and compact operators on the Bergman space [formula omitted] in the unit ball of [formula omitted

Let μ be a complex Borel measure on the unit ball of C n and α > − 1 . We characterize the measures μ for which the Toeplitz operator T μ α is bounded or compact on the Bergman space L a 1 ( B n , ( 1 − | z | 2 ) α d ν ) , where dν is the normalized Lebesgue measure on the unit ball of C n . Our results also include the case of more general operators in L a 1 ( B n , d ν ) . These results extend to several dimensions the results of Agbor, Békollé and Tchoundja (2011) [2] and Wu, Zhao and Zorborska (2006) [11].

Law of the Sum of Bernoulli Random Variables

Let $\Delta_n$ be the set of all possible joint distributions of n Bernoulli random variables $X_{1},\ldots,X_n$. Suppose that $\Delta_n$, which is a simplex in the $2^n$-dimensional space, is endowed with the normalized Lebesgue measure $\mu_n$. Suppose also that the integer n is large. Then we show that there is a subset $\Delta $ of $\Delta_n$, whose measure $\mu_n(\Delta )$ is very close to 1, such that if the joint distribution of $(X_{1},\ldots,X_n)$ is in $\Delta $, then the law of the sum $X_{1}+\cdots+X_n$ is close to the binomial law ${\cal B}(n,\frac12)$. This result does not need any independence assumption. Next, we show a result of the same kind when $\Delta_n$ is endowed with another probability measure $\nu_n$.

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

Let D ⊂ R d D\subset R^d be a bounded domain and let P ( D ) \mathcal P(D) denote the space of probability measures on D D . Consider a Brownian motion in D D which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity γ &gt; 0 \gamma &gt;0 to a new point, according to a distribution μ ∈ P ( D ) \mu \in \mathcal P(D) . From this new point it repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator − L γ , μ -L_{\gamma ,\mu } , defined by L γ , μ u ≡ − 1 2 Δ u + γ V μ ( u ) , \begin{equation*} L_{\gamma ,\mu }u\equiv -\frac 12\Delta u+\gamma V_\mu (u), \end{equation*} with the Dirichlet boundary condition, where V μ V_\mu is a nonlocal “ μ \mu -centering” potential defined by V μ ( u ) = u − ∫ D u d μ . \begin{equation*} V_\mu (u)=u-\int _Du d\mu . \end{equation*} The operator L γ , μ L_{\gamma ,\mu } is symmetric only in the case that μ \mu is normalized Lebesgue measure; thus, only in that case can it be realized as a selfadjoint operator. The corresponding semigroup is compact, and thus the spectrum of L γ , μ L_{\gamma ,\mu } consists exclusively of eigenvalues. As is well known, the principal eigenvalue gives the exponential rate of decay in t t of the probability of not exiting the domain by time t t . We study the behavior of the eigenvalues, our main focus being on the behavior of the principal eigenvalue for the regimes γ ≫ 1 \gamma \gg 1 and γ ≪ 1 \gamma \ll 1 . We also consider conditions on μ \mu that guarantee that the principal eigenvalue is monotone increasing or decreasing in γ \gamma .

Invariant densities for random $\beta$-expansions

Let \beta &gt;1 be a non-integer. We consider expansions of the form \sum_{i=1}^{\infty} \frac{d_i}{\beta^i} , where the digits (d_i)_{i \geq 1} are generated by means of a Borel map K_{\beta} defined on \{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right] . We show existence and uniqueness of an absolutely continuous K_{\beta} -invariant probability measure w.r.t. m_p \otimes \lambda , where m_p is the Bernoulli measure on \{0,1\}^{\N} with parameter p ( 0 &lt; p &lt; 1) and \lambda is the normalized Lebesgue measure on [0 ,\lfloor \beta \rfloor /(\beta -1)] . Furthermore, this measure is of the form m_p \otimes \mu_{\beta,p} , where \mu_{\beta,p} is equivalent with \lambda . We establish the fact that the measure of maximal entropy and m_p \otimes \lambda are mutually singular. In case 1 has a finite greedy expansion with positive coefficients, the measure m_p \otimes \mu_{\beta,p} is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].