Fourier asymptotics of statistically self-similar measures

In this paper we investigate the pointwise Fourier decay of some selfsimilar random measures. As an application we construct statistically selfsimilar Salem sets. For example, our result shows that a slight random perturbation of the classical Cantor set becomes a nice set in the sense that its Fourier dimension equals its Hausdorff dimension.     

On a relationship between the integrabilities of various maximal functions

It is shown that the right-sided, left-sided, and symmetric maximal functions of any measurable function can be integrable only simultaneously. The analogous statement is proved for the ergodic maximal functions.     

On the Beurling dimension of exponential frames

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff dimension of the fractal.     

Fourier transforms of irregular mixed homogeneous hypersurface measures

With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems.     

A fixed point theorem for moment matrices of self-similar measures

We consider the self-similar measure on the complex plane C$\mathbb{C}$ associated to an iterated function system (IFS) with probabilities. From this IFS we define an operator in a complete metric space of infinite matrices. Using the expression obtained in a previous work of the authors, we prove that this operator has as fixed point the moment matrix of the self-similar measure. As a consequence, we obtain a very efficient algorithm to compute the moment matrix of the self-similar measure. Finally, in order to estimate the rate of convergence of the algorithm, we find an upper bound of the norm of this contractive operator.     

The convergence inL 1 of singular integrals in harmonic analysis and ergodic theory

We study the behavior of the ergodic singular integral associated to a nonsingular measurable flow {:t } on a finite measure space and a Calderón—Zygmund kernel with support in (0, ). We show that if the flow preserves the measure or, with more generality, if the flow is such that the semiflow {_t:t>-0} is Cesàrobounded,f and f are integrable functions, then the truncations of the singular integral converge to f not only in the a.e. sense but also in the L¹-norm. To obtain this result we study the problem for the singular integrals in the real line and in the setting of the weighted L¹-spaces.This research has been partially supported by a D.G.I.C.Y.T. grant (PB94-1496), a D.G.E.S. grant (PB97-1097) and Junta de Andalucía.     

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG ⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: GC(S ^n–1) and a kernelK: ((G × G }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with (E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.     

Fourier Multipliers and Littlewood‐Paley for modulation spaces

In this paper we have studied Fourier multipliers and Littlewood‐Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space into itself possesses an l₂‐valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical ‐spaces.     

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in {0,1,2,3}) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.     

Pointwise Behavior of Double Fourier Series of Functions of Bounded Variation

We extend some recent results of S. A. Telyakovskii on the uniform boundedness of the partial sums of Fourier series of functions of bounded variation to periodic functions of two variables, which are of bounded variation in the sense of Hardy. As corollaries, we obtain the classical Parseval formula, the convergence theorem of the series involving the sine Fourier coefficients, and a lower estimate of the best approximation by trigonometric polynomials in the metric of L in a sharpened version. This research was supported by the Hungarian National Foundation for Scientific Research under Grants TS 044 782 and T 046 192.     

Inhomogeneous discrete calderón reproducing formulas for spaces of homogeneous type

In this article a Littlewood-Paley theorem for a new kind of Littlewood-Paley g-functions over spaces of homogeneous type is presented. Based on it the authors establish inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type, making use of Calderón-Zygmund operators.     

Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation

Sharp inequalities between weight bounds (from the doubling, A_p, and reverse Hölder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A_∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Hölder weights and extend Coifman and Fefferman's formulation of the A_∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.     

Beurling dimension and self-similar measures

In this paper the authors study the Beurling dimension of Bessel sets and frame spectra of some self-similar measures on ${\mathbb{R}}^d$ and obtain their exact upper bound of the dimensions, which is the same given by Dutkay et al. (2011) [8]. The upper bound is attained in usual cases and some examples are given to explain our theory.     

Duality relation on spectra of self‐affine measures

The present paper establishes a duality relation for the spectra of self‐affine measures. This is done under the condition of compatible pair and is motivated by a duality conjecture of Dutkay and Jorgensen on the spectrality of self‐affine measures. For the spectral self‐affine measure, we first obtain a structural property of spectra which indicates that one can get new spectra from old ones. We then establish a duality property for the spectra which confirms the conjecture in a certain case.     

Topological bounds for Fourier coefficients and applications to torsion

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded convex domain in the plane and consider

$\begin{array}{cc}\hfill ?\mathrm{\Delta }u& =1\text{in}\mathrm{\Omega }\hfill \\ \hfill u& =0\text{on}?\mathrm{\Omega }.\hfill \end{array}$

If u assumes its maximum in $x_0\in \mathrm{\Omega }$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^{2}u(x_0)$. We prove that $D^{2}u(x_0)$ is negative definite and give a quantitative bound on the spectral gap

$\begin{array}{c}{\lambda }_{\max }(D^{2}u(x_0))\le ?c_1\exp ?(?c_2\frac{\mathrm{diam}(\mathrm{\Omega })}{\mathrm{inrad}(\mathrm{\Omega })})\hfill \\ \text{for universal}{c}_1,c_2>0.\hfill \end{array}$

This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if $f:\mathbb{T}\to \mathbb{R}$ is continuous and has n sign changes, then

$\sum _{k=0}^{n/2}\left|〈f,\sin ?kx〉\right|+\left|〈f,\cos ?kx〉\right|{?}_n\frac{{|f6}_{L^1(\mathbb{T})}^{n+1}}{{6f6}_{L^{\infty }(\mathbb{T})}^n}.$

This statement immediately implies estimates on higher derivatives of harmonic functions u in the unit ball: if u is very flat in the origin, then the boundary function $u(\cos ?t,\sin ?t):\mathbb{T}\to \mathbb{R}$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:\mathbb{T}\to \mathbb{R}$ cannot decay faster than $～\exp ?(?{(\mathrm{\#}\text{sign changes})}^{2}t/4)$.     

Lévy group and density measures

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group G of permutations of N. Using a new characterization of the Lévy group G we will prove that a finitely additive measure extends density if and only if it is G-invariant.     

On the stability of wavelet and Gabor frames (Riesz bases)

If the sequence of functions _{j, k} is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system _j,k which is still a frame (Riesz basis) under very mild conditions. For example, we do not need to know that the support of or is compact as in [14]. We also discuss the stability of irregular sampling problems. In order to arrive at some of our results, we set up a general multivariate version of Littlewood-Paley type inequality which was originally considered by Lemarié and Meyer [17], then by Chui and Shi [9], and Long [16].     

Scaling of spectra of a class of self‐similar measures on R

Let be two positive integers. For , let the self‐similar measure be defined by . It is known [18] that is a spectral measure with a spectrum where . In this paper, we give some conditions on under which the scaling set is also a spectrum of .