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This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems ; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

In this paper, we obtain explicit branching laws for all irreducible unitary representations of G=\operatorname {Spin}(N,1) when restricted to a parabolic subgroup P. The restriction turns out to be a finite direct sum of irreducible unitary representations of P. We also verify Duflo's conjecture for the branching laws of discrete series representations of G with respect to P. That is to show: in the framework of the orbit method, the branching law of a discrete series representation is determined by some geometric behavior of the moment map for the corresponding coadjoint orbit. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group and let K be its closed subgroup. Write \widehat {G}_{K} for the set of irreducible representations with non-zero K-invariant vectors. We call a pair (G,K) admissible if for each irreducible representation (\pi, V_{\pi }) in \widehat {G}_{K}, its K-invariant subspace V_{\pi }^{K} is of finite dimension. For each \pi in \widehat {G}_{K}, let \pi _{v_{i}, \overline {\xi }_{j}}'s (\pi _{v_{i}, \overline {\xi }_{j}}(gK)≔\langle v_{i}, \pi (g)\xi _{j}\rangle) be the matrix coefficeints on G/K induced by fixed orthonormal bases \{v_{i}\} and \{\xi _{j}\} for V_{\pi } and V_{\pi }^{K} respectively. A probability measure \mu on G/K is called a spectral measure if there is a subset \Gamma of \widehat {G}_{K} such that the set of all such matrix coefficients \pi _{v_{i}, \overline {\xi }_{j}},\ \pi \in \Gamma, constitutes an orthonormal basis for L^{2}(G/K, \mu) with some suitable normalization of these matrix coordinate functions. In this paper, we shall give a characterization of a spectral measure for an admissible pair (G,K) by using the Fourier transform on G/K. Also, from this we show that there is a "local translation" (we call it locally regular representation in the sequel) of G on L^{2}(G/K, \mu) under a mild condition. This will give us some necessary conditions for the existence of spectral measures. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied. [ABSTRACT FROM AUTHOR]

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403-2425]. [ABSTRACT FROM AUTHOR]

This paper uses exponential sum methods to detect the presence of k-chains of points whose endpoints belong to proper subsets of finite p-adic rings. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of global ultradifferentiable functions with respect to weight functions and include a discussion of its functional analytic theory and prove a characterization in terms of certain exponential decay of the Fourier-Broz-Iagolnitzer transform—a Paley-Wiener type theorem. As an application we investigate the regularity of global ultradifferentiable vectors proving a version of the Kotake-Narasimhan theorem in this setting. [ABSTRACT FROM AUTHOR]

This paper studies averages of finite point configuration problems for subsets E ⊂ (Z/pr)n (r ≥ 1, n ≥ 2) and extends work of Bennett-Hart-Iosevich-Pakianathan-Rudnev over finite fields to finite p-adic rings. As a result, we show that averages, taken over the group of orthogonal transformations, of finite point configurations with endpoints in E are positive if the density of E is sufficiently large. [ABSTRACT FROM AUTHOR]

In this paper, we follow an idea of Lusztig to define the Fourier transform on the Ringel-Hall algebra of a valued quiver (given by a quiver with automorphism). As an application, this provides a direct proof of the fact that the Ringel-Hall algebra of a valued quiver is independent of its orientation. Furthermore, by combining the BGP-reflection operators defined on double Ringel-Hall algebras of valued quivers with Fourier transforms, we obtain an alternative construction of Lusztig's symmetries of the associated quantum enveloping algebras. This generalizes a result of Sevenhant and Van den Bergh in the quiver case. [ABSTRACT FROM AUTHOR]

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho-Stark uncertainty principle, and Meshulam's nonabelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator L1 → L∞, and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new proofs and many generalizations of these basic uncertainty principles, to new operators and to new settings, in a completely unified way. Together with our general overview, this paper can also serve as a survey of the many facets of the phenomena known as uncertainty principles. [ABSTRACT FROM AUTHOR]

David Goss developed a very general Fourier transform in additive harmonic analysis in the function field setting. In order to introduce the Fourier transform for continuous characteristic p-valued functions on Zp, Goss introduced and studied an analogue of flows in finite characteristic. In this paper, we use another approach to study flows in finite characteristic. We recast the notion of a flow in the language of the classical umbral calculus, which allows us to generalize the formula for flows first proved by Goss to a more general setting. We study duality between flows using the classical umbral calculus, and show that the duality notion introduced by Goss seems to be a natural one. We also formulate a question of Goss about the exact relationship between two flows of a Drinfeld module in the language of the classical umbral calculus, and give a partial answer to it. [ABSTRACT FROM AUTHOR]

The quaternion Fourier transforms are powerful tools in modern data analysis, in particular for color image processing. At present, there are mainly three different quaternion Fourier transforms widely used. In this paper, we prove the sharp Hausdorff-Young inequalities for these three transforms and the more general ones, i.e., the steerable quaternion Fourier transforms. Then Hirschman's entropy uncertainty principle in the quaternion setting follows from the standard differential approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of the SYZ transform on a semiflat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and the extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multisections defined by Arinkin and Joyce [Fukaya category and Fourier transform, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2001] and Leung, Yau, and Zaslow [Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319-1341]. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold (M, ω), under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho and Joyce [J. Differential Geom. 86 (2010), no. 3, 831-500, Question 13.15]. Furthermore, if M admits a Lagrangian torus fibration over an integral affine manifold, we prove, under some additional assumptions, that this new equivalence is mirror to an isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform. [ABSTRACT FROM AUTHOR]

We consider the following semirelativistic nonlinear Schr¨odinger equation (SNLS): { i∂tu ± (m² - Δ)1/2u = λ|u|p, (t, x) ∈ [0, T) x ℝd, u(0, x) = u0(x), x ∈ ℝd, where m ≥ 0, λ ∈ C \ {0}, d ∈ N, T > 0, and ∂t = ∂/∂t. Here (m² -Δ)1/2 := F-1(m² + |ξ|²)1/2F, where F denotes the Fourier transform. Fujiwara and Ozawa proved the nonexistence of global weak solutions to SNLS for some initial data in the case of d = 1, m = 0, and 1 < p ≤ 2 by a test function method. In this paper, we extend their result to a more general setting: for example, m ≥ 0, d ∈ N, or p > 1. Moreover, we obtain the upper estimates of weak solutions to SNLS. The key to the proof is to choose an appropriate test function. [ABSTRACT FROM AUTHOR]

In this paper, we show that non-symmetric convex polytopes cannot serve as a window function to produce a Gabor orthonormal basis by any time-frequency sets. [ABSTRACT FROM AUTHOR]

Schoenberg (1938) identified the class of positive definite radial (orisotropic) functions φ : R...d → R, φ(0) φ(0) = 1, as having a representation φ(x) = R+Ωd(tu)Gd(du), t = ∥x∥, for some uniquely identified probability measure Gd on R...+ and Ωd(t) = E(eit≺e1,η≻), where η is a vector uniformly distributed on the unit spherical shell Sd-1 ⊂ R...d and e1 is a fixed unit vector. Call such Gd a d-Schoenberg measure, and let Φd denote the class of all functionsf : R...+ → R for which such a d-dimensional radial function Φ exists with f(t) = φ(x) for t = ∥x∥. Mathéron (1965) introduced operators I... and D..., called Montée and Descente, that map suitable f ∈ Φd into Φd for some different dimension d': Wendland described such mappings as dimension walks. This paper characterizes Mathéron's operators in terms of Schoenberg measures and describes functions, even in the class Φ∞ of completely monotone functions, for which neither I...f nor D...f is well defined. Because f ∈ Φd implies f ∈ Φd' for d' < d, any f ∈ Φd has a d'-Schoenberg measure Gd' for 1 ≤ d' < d and finite d ≥ 2. This paper identifies Gd' in terms of Gd via another 'dimension walk' relating the Fourier transforms Ωd' and Ωd that reflect projections on R...d' within R...d. A study of the Euclid hat function shows the indecomposability of Ωd. [ABSTRACT FROM AUTHOR]

Let K be a quadratic number field and let E be a Q-curve without CM completely defined over K and not isogenous to an elliptic curve over Q. In this setting, it is known that there exists a weight 2 newform of suitable level and character, such that L(E, s) = L(f, s)L(σf, s), where σf is the unique Galois conjugate of f. In this paper, we first describe an algorithm to compute the level, the character and the Fourier coefficients of f. Next, we show that given an invariant differential ωE on E, there exists a positive integer ℚ = ℚ(E, ωE) such that L(E, 1)/P(E/K·ℚ is an integer, where P(E/K) is the period of E. Assuming a generalization of Manin's conjecture, the integer ℚ is made effective. As an application, we verify the weak BSD conjecture for some curves of rank two, we compute the L-ratio of a curve of rank zero and we produce relevant examples of newforms of large level. [ABSTRACT FROM AUTHOR]

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives near the critical line. As an application, we compute the one-level density for test functions whose Fourier transform is supported in \left (-2,2\right), including lower-order terms. [ABSTRACT FROM AUTHOR]

It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on \mathbb R^n, modified by a weight, then the weight is bounded above and below and the space is equivalent to L^2. Also, if it is bounded from L^p to L^q, each modified by the same weight, then the weight is bounded above and below and 1\le p=q'\le 2. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation. [ABSTRACT FROM AUTHOR]

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. Motivated by the paper [H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996) 887- 898], in this paper we introduce two new classes of orthogonal functions, which are respectively Fourier transforms of the generalized ultraspherical polynomials and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval's identity. [ABSTRACT FROM AUTHOR]

This paper is devoted to designing some modified composite Gauss type rules for the integrals involving algebraic and logarithmic endpoint singularities, at which the integrands possess the Puiseux expansions in series. Firstly, the error asymptotic expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sum of the Hurwitz zeta function and its higher derivatives. Secondly, the deduced error asymptotic expansion is applied to the composite Gauss-Legendre and Gauss-Kronrod rules. By simplifying the evaluations of the Hurwitz zeta function and its derivatives, two modified composite Gaussian rules and their error estimates are obtained. The methods can also effectively deal with infinite range singular integrals, the singular and oscillatory Fourier transforms and the Cauchy principal value integrals by simple variable transformations. The advantage of the practical algorithms is that three ways can be used to increase the accuracy of the algorithms, which are decreasing the step length of the composite rule, increasing the orders of the Puiseux expansions and increasing the number of nodes of the Gaussian quadrature rules. Finally, the excellent performance of the proposed methods is demonstrated through several typical numerical examples. It is shown that the algorithms can be used to automatically evaluate a wide range of singular integrals over finite or infinite intervals. [ABSTRACT FROM AUTHOR]

We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of the z-axis and if dS denotes the surface measure for this surface, then the measures under consideration are of the form K(x, y)g(z) dS where K(x, y)g(z) is supported near the origin and both K(x, y) and g(z) are allowed to have singularities. The estimates here generalize the previously known sharp uniform estimates for when K(x, y)g(z) is smooth. The methods used in this paper involve an explicit two-dimensional resolution of singularities theorem, iterated twice, coupled with Van der Corput-type lemmas. [ABSTRACT FROM AUTHOR]

This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroup-adapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easy-to-use upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the best known fast transforms for some abelian groups, the symmetric groups and their wreath products, and the classical Weyl groups. Beyond this, we obtain greatly improved upper bounds for the general linear and unitary groups over a finite field, and for the classical Chevalley groups over a finite field. We also apply these techniques to obtain analogous results for homogeneous spaces. This is part I of a two part paper. Part II will present a refinement of these techniques which yields further savings. [ABSTRACT FROM AUTHOR]

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems. Our first main result considers the orthogonal projections of two Borel sets A,B ⊂ R2 into one-dimensional subspaces. Under the assumptions dim A ⩽ 1 < dim B and dim A + dim B > 2, we prove that the intersection of the projections PL(A) and PL(B) has dimension at least dim A - ϵ for positively many lines L, and for any ϵ > 0. This is quite sharp: given s, t ∊ [0, 2] with s+t = 2, we construct compact sets A,B ⊂ R2 with dim A = s and dim B = t such that almost all intersections PL(A) ∩ PL(B) are empty. In case both dimA > 1 and dimB > 1, we prove that the intersections PL(A) ∩ PL(B) have positive length for positively many L. If A ⊂ R2 is a Borel set with 0 < Hs(A) < ∝ for some s > 1, it is known that A is "visible" from almost all points x ∊ R2 in the sense that A intersects a positive fraction of all lines passing through x. In fact, a result of Marstrand says that such non-empty intersections typically have dimension s - 1. Our second main result strengthens this by showing that the set of exceptional points x ∊ R2, for which Marstrand's assertion fails, has Hausdorff dimension at most one. [ABSTRACT FROM AUTHOR]

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1∣2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator. [ABSTRACT FROM AUTHOR]

We present three algorithms to calculate Φn(z), the nth cyclotomic polynomial. The first algorithm calculates Φn(z) by a series of polynomial divisions, which we perform using the fast Fourier transform. The second algorithm calculates Φn(z) as a quotient of products of sparse power series. These two algorithms, described in detail in the paper, were used to calculate cyclotomic polynomials of large height and length. In particular, we have found the least n for which the height of Φn(z) is greater than n, n2, n3, and n4, respectively. The third algorithm, the big prime algorithm, generates the terms of Φn(z) sequentially, in a manner which reduces the memory cost. We use the big prime algorithm to find the minimal known height of cyclotomic polynomials of order five. We include these results as well as other examples of cyclotomic polynomials of unusually large height, and bounds on the coefficient of the term of degree k for all cyclotomic polynomials. [ABSTRACT FROM AUTHOR]

We study upper bounds, approximations, and limits for functions of motivic exponential class uniformly in non-Archimedean local fields whose characteristic is 0 or sufficiently large. Our results together form a flexible framework for doing analysis over local fields in a field-independent way. As corollaries, we obtain many new transfer principles, for example, for local constancy, continuity, and existence of various kinds of limits. Moreover, we show that the Fourier transform of an L2-function of motivic exponential class is again of motivic exponential class. As an application in representation theory, we prove uniform bounds for the Fourier transforms of orbital integrals on connected reductive p-adic groups. [ABSTRACT FROM AUTHOR]

The L^p boundedness theory of convolution operators is based on an initial L^2\to L^2 estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we obtain initial L^2\times \cdots \times L^2\to L^{2/m} estimates for three types of important multilinear operators: rough singular integrals, multipliers of Hörmander type, and multipliers whose derivatives satisfy qualitative estimates. These estimates lay the foundation for the derivation of other L^p estimates for such operators. [ABSTRACT FROM AUTHOR]

We construct a function that lies in L^p(\mathbb {R}^d) for every p \in (1,\infty ] and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač's maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set. [ABSTRACT FROM AUTHOR]

We use the classical Fourier analysis on \mathbb {R}^n to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained for totally geodesic Funk transforms on the sphere and the relevant \lambda-cosine transforms. [ABSTRACT FROM AUTHOR]

We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to nongroup semigroups. [ABSTRACT FROM AUTHOR]

In the present paper, the dynamic behavior of functionally graded piezoelectric materials is investigated when it is under anti-plane mechanical loading and in-plane electrical loading. It is assumed that the shear modulus, the piezoelectric modulus, the dielectric modulus and mass density of FGPM vary continuously as functions of X and Y. By using Fourier transforms the solution of equilibrium equations is obtained in closed form. The expressions for displacement and electrical potential are obtained in terms of one unknown function. Finally the results are applied to obtain a solution of the moving contact problem on the surface of the functionally graded piezoelectric material (FGFM). [ABSTRACT FROM AUTHOR]

Lp-Lq decay estimates of solutions to the Cauchy problem of linear thermoelastic systems with second sound in one space variable will be studied in this paper. First, by dividing the frequency of phase space of the Fourier transformation into different regions, the asymptotic behavior of characteristic roots of the coefficient matrix is obtained by carefully analyzing the effect of the different regions. Second, with the help of the information on the characteristic roots and by using the interpolation theorem, the Lp-Lq decay estimate of solutions to the Cauchy problem of the linear thermoelastic system with second sound in one space variable is obtained. [ABSTRACT FROM AUTHOR]

Let $B$ be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of $B$, and the consequences of these properties for the distribution of lattice points in rotated and translated copies of $\rho B$ ($\rho$ being a large positive number), irregularities of distribution, and the spherical average decay of the Fourier transform of the characteristic function of $B$. The analysis makes use of two notions of ``dimension'' of a convex set. The first notion is defined in terms of the number of sides required to approximate a convex set by a polygon up to a certain degree of accuracy. The second is the fractal dimension of the image of the Gauss map of $B$. The results stated in terms of these quantities are essentially sharp and lead to a nearly complete description of the problems in question. [ABSTRACT FROM AUTHOR]

Hardy's uncertainty principle states that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. In this paper we prove $L^p$ versions of this principle for the Jacobi transform and for the Fourier transform on real hyperbolic spaces. [ABSTRACT FROM AUTHOR]

In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in $\mathbb R^2$ with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given. [ABSTRACT FROM AUTHOR]

In this paper we introduce new function spaces that are denoted by ${\mathcal H}_{\mu ,p}$, $\mu >-1/2$ and $1\leq p\leq \infty ,$ and that are spaces of ${\mathcal D}_{L^{p}}-$type where the Hankel convolution and the Hankel transformation are defined. The spaces ${\mathcal H}_{\mu ,p}$ will play the same role in the Hankel setting that the spaces ${\mathcal D}_{L^{p}}$ play in the theory of Fourier transformation. [ABSTRACT FROM AUTHOR]

In this paper we define a generalized finite Fourier transformation in distribution spaces. Also we investigate a distributional convolution for this finite integral transformation. [ABSTRACT FROM AUTHOR]

We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erdős and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure. [ABSTRACT FROM AUTHOR]

The Hardy uncertainty principle says that no function is better localized together with its Fourier transform than the Gaussian. The textbook proof of the result, as well as one of the original proofs by Hardy, refers to the Phragmén–Lindelöf theorem. In this note we first describe the connection of the Hardy uncertainty to the Schrödinger equation, and give a new proof of Hardy's result which is based on this connection and the Liouville theorem. The proof is related to the second proof of Hardy, which has been undeservedly forgotten. Then we survey the recent results on dynamical versions of Hardy's theorem. [ABSTRACT FROM AUTHOR]

By using the Fourier transform, we successfully give Kröger-type estimates for sums of eigenvalues of the free plate (under tension and with nonzero Poisson's ratio) in terms of the dimension of the ambient space, the volume of the domain, the tension parameter and the Poisson's ratio. [ABSTRACT FROM AUTHOR]

Let $A$ be a Banach algebra with a bounded approximate identity. Let $Z_1$ and $\widetilde Z_2$ be, respectively, the topological centers of the algebras $A^{**}$ and $(A^*A)^*$. In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras $L^1(G)$ and $A(G)$, we study the sets $Z_1$, $\widetilde Z_2$, the relations between them and with several other subspaces of $A^{**}$ or $A^*$. [ABSTRACT FROM AUTHOR]

An uncertainty principle, due to Hardy, for Fourier transform pairs on $\Bbb R$ says that if the function $f$ is ``very rapidly decreasing'', then the Fourier transform cannot also be ``very rapidly decreasing'' unless $f$ is identically zero. In this paper we state and prove an analogue of Hardy's theorem for the $n$-dimensional Euclidean motion group. [ABSTRACT FROM AUTHOR]

The existence of the `rare' sequence of partial sums summable with the method of arithmetical means at each Lebesgue point is proved in the paper. The proof is based on the strategy of random choice. [ABSTRACT FROM AUTHOR]

Let μ be an α-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform μ. More precisely, if H is a truncated hyperbolic paraboloid in Rd we study the optimal β for which ∫H |μ(R\xi)|2, d σ (ξ) ≤ C(α, μ) R−β for all R > 1. Our estimates for β depend on the minimum between the number of positive and negative principal curvatures of H; if this number is as large as possible our estimates are sharp in all dimensions. [ABSTRACT FROM AUTHOR]

We discuss the structural and topological properties of a general class of weighted L1 convolutor spaces. Our theory simultaneously applies to weighted D'L1 spaces as well as to convolutor spaces of the Gelfand-Shilov spaces K\Mp. In particular, we characterize the sequences of weight functions (Mp)p ∈ N for which the space of convolutors of K\Mp is ultrabornological, thereby generalizing Grothendieck's classical result for the space O'C of rapidly decreasing distributions. Our methods lead to the first direct proof of the completeness of the space OC of very slowly increasing smooth functions. [ABSTRACT FROM AUTHOR]

We consider a reconstruction problem for ''spike-train'' signals F of an a priori known form F(x) = ∑j = 1dajδ(x−xj), from their moments mk(F) = ∫ xkF(x)dx. We assume that the moments mk(F), k = 0,1, . . . ,2d−1, are known with an absolute error not exceeding ε > 0. This problem is essentially equivalent to solving the Prony system ∑j = 1d ajxjk = mk(F), k = 0,1, . . . ,2d−1. We study the ''geometry of error amplification'' in reconstruction of F from mk(F), in situations where the nodes x1, . . . ,xd near-collide, i.e., form a cluster of size h << 1. We show that in this case, error amplification is governed by certain algebraic varieties in the parameter space of signals F, which we call the ''Prony varieties''. Based on this we produce lower and upper bounds, of the same order, on the worst case reconstruction error. In addition we derive separate lower and upper bounds on the reconstruction of the amplitudes and the nodes. Finally we discuss how to use the geometry of the Prony varieties to improve the reconstruction accuracy given additional a priori information. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness of L2 solutions to the inhomogeneous wave equation on Rn × R under the assumption that the inhomogeneous data lies in Lp(Rn+1) for p = (2n + 2)/(n + 5) and n + 1 ≥ 5. We also require the Fourier transform of the inhomogeneous data to vanish on an infinite cone where the solution could become singular. Subsequently, we show sharpness of the exponent p. This extends work of Michael Goldberg, in which similar Fourier-analytic techniques were used to study the inhomogeneous Helmholtz equation. [ABSTRACT FROM AUTHOR]

One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform Hƒ(z) of ƒ for z ∈ C has only real zeros when ƒ(t) is a specific function Φ (t). Pólya's 1920s approach to the RH extended Hƒto Hƒ,λ, the Fourier transform of eλt2 ƒ(t). We review developments of this approach to the RH and related ones in statistical physics where ƒ(t) is replaced by a measure d ρ (t). Pólya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant ΛDN = ΛDN (Φ) in (−∞ 1/2] such that HΦ,λ has only real zeros if and only if λ ≥ ΛDN; the RH is then equivalent to ΛDN ≤ 0. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that ΛDN ≥ 0 (that the RH, if true, is only barely so) and the Polymath 15 project improving the 1/2 upper bound to about 0.22. We also present examples of ρ's with differing Hρ,λ and ΛDN (ρ) behaviors; some of these are new and based on a recent weak convergence theorem of the authors. [ABSTRACT FROM AUTHOR]

We consider several problems related to the restriction of (∇k) ƒ to a surface Σ ⊂ Rd with nonvanishing Gauss curvature. While such restrictions clearly exist if ƒ is a Schwartz function, there are few bounds available that enable one to take limits with respect to the Lp(Rd) norm of ƒ. We establish three scenarios where it is possible to do so: When the restriction is measured according to a Sobolev space H−s(Σ) of negative index, we determine the complete range of indices (k, s, p) for which such a bound exists. Among functions where ƒ vanishes on Σ to order k−1, the restriction of (∇k) ƒ defines a bounded operator from (this subspace of) Lp(Rd) to L2(Σ) provided 1 ≤ p ≤ 2d+2/d+3+4k. When there is a priori control of ƒ||Σ in a space Hl(Σ), l > 0, this implies improved regularity for the restrictions of (∇k)ƒ. If l is large enough, then even ||∇ƒ||L2(Σ) can be controlled in terms of ||ƒ||{Hl(Σ) and ||ƒ||Lp(Rd) alone. The proofs are based on three main tools: the spectral synthesis work of Y.Domar, which provides a mechanism for Lp approximation by ''convolving along surfaces in spectrum'', a new bilinear oscillatory integral estimate valid for ordinary Lp functions, and a convexity-type property of the quantity ||(∇k) ƒ||H−s(Σ)} as a function of k and s that allows one to employ the control of ||ƒ||Hl (Σ). [ABSTRACT FROM AUTHOR]

We consider a one-dimensional logistic like equation with a nonlinear and nonlocal diffusion term with periodic boundary conditions. In this note we focus on smooth nonlocal kernels exhibiting a repulsive effect, that is expressed in terms of the positivity of their Fourier transforms. Here we describe the large time behaviour of the solutions for a large class of initial data. We roughly prove that the nontrivial solutions converge strongly to the unique spatially homogeneous equilibrium, by providing a refined study of the properties of the characteristic curves associated to the problem. [ABSTRACT FROM AUTHOR]