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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]

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]

For a finite positive Borel measure μ on R its exponential type, T μ , is defined as the infimum of a > 0 such that finite linear combinations of complex exponentials with frequencies between 0 and a are dense in L 2 (μ). The definition can be easily extended from finite to broader classes of measures. In this paper we prove a new formula for T μ and use it to study growth and additivity properties of measures with finite positive type. As one of the applications, we show that Frostman measures on R may only have type zero or infinity. [ABSTRACT FROM AUTHOR]

Abstract Let M be a holonomic algebraic D -module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to M at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of M are obtained by Legendre transform from the exponential factors of M. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann–Hilbert correspondence. [ABSTRACT FROM AUTHOR]

In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R 2 so that ( Γ , Λ ) becomes a HUP. [ABSTRACT FROM AUTHOR]

A Heisenberg uniqueness pair is a pair ( Γ , Λ ) , where Γ is a curve and Λ is a set in R 2 such that whenever a finite Borel measure μ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfies μ ˆ | Λ = 0 , then it is identically 0. In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials. [ABSTRACT FROM AUTHOR]

In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing complex and quaternionic convex bodies. We conduct a systematic study of sections of bodies with symmetries of this type, with the emphasis on problems of the Busemann–Petty type and hyperplane inequalities. The main role belongs to the class of intersection bodies with symmetries. [ABSTRACT FROM AUTHOR]

We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain–Katz–Tao finite field incidence theorem derived from sum-product estimates. In prime order fields, we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid. [ABSTRACT FROM AUTHOR]

Abstract: For and an integer, let μ be the self-similar measure defined by . We prove that has an exponential orthonormal basis if and only if for some and N divides q. The special case is the Cantor measure with and [16], which was proved recently to be the only spectral measure among the Bernoulli convolutions with [4]. [Copyright &y& Elsevier]