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A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, α -mixing and τ -mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. More precisely, the estimator is able to reach the convergence rate induced by the best Sobolev regularity of the density of A X , where A belongs to a class of invertible matrices describing all the possible directions. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas. [ABSTRACT FROM AUTHOR]

Let { S d } d = 0 2 N − 1 be an iterated function system defined by S d (x) = (− 1) d ρ (x + d) , x ∈ R , where 0 < ρ < 1 and N is a positive integer. Let μ ρ , D 2 N be a self-similar measure defined by μ ρ , D 2 N (⋅) = 1 2 N ∑ d = 0 2 N − 1 μ ρ , D 2 N (S d − 1 (⋅)). We show that μ ρ , D 2 N is a spectral measure if and only if ρ = p − 1 for some nature number p and 2 N divides p. Moreover, we introduce a useful way to determine the spectrality of the convolution of two infinitely supported measures. [ABSTRACT FROM AUTHOR]

The paper concerns problems of the recovery of operators from noisy information in weighted L q -spaces with homogeneous weights. A number of general theorems are proved and applied to finding exact constants in multidimensional Carlson type inequalities with several weights and problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of generalized Laplace operators from a noisy Fourier transform in the L p -metric. [ABSTRACT FROM AUTHOR]

An infinitely smooth symmetric convex body K ⊂ R d is called k -separably integrable, 1 ≤ k < d , if its k -dimensional isotropic volume function V K , H (t) = H d ({ x ∈ K : dist (x , H ⊥) ≤ t }) can be written as a finite sum of products in which the dependence on H ∈ Gr k (R d) and t ∈ R is separated. In this paper, we will obtain a complete classification of such bodies. Namely, we will prove that if d − k is even, then K is an ellipsoid, and if d − k is odd, then K is a Euclidean ball. This generalizes the recent classification of polynomially integrable convex bodies in the symmetric case. [ABSTRACT FROM AUTHOR]

Let F q [ t ] be the polynomial ring over the finite field F q of q elements. For 1 ≤ j ≤ l , let a j , a j ′ ∈ F q [ t ] with a j ≠ 0. Let H (x) denote the product of l linear functions a j x + a j ′. Let X (H) be the set of all monic polynomials n in F q [ t ] such that H (n) is the product of l monic irreducible polynomials in F q [ t ]. The Hardy-Littlewood l -tuple conjecture provides an asymptotic formula for the cardinality of the set A N : = { n ∈ X (H) : deg ⁡ n = N } when N tends to ∞. In this paper an F q [ t ] -version of the enveloping sieve theory is established by following the constructing method of Green-Tao. It is applied to give an upper bound for the cardinality of ⋃ j < N A j. It is also applied to obtain Lebesgue space estimates for the exponential sum over X (H). These results are the best possible if the Hardy-Littlewood l -tuple conjecture is true. [ABSTRACT FROM AUTHOR]