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Abstract Let L j = ∂ t j + (a j + i b j) (t j) ∂ x , j = 1 , ... , n , be a system of vector fields defined on the torus T t n × T x 1 , where the coefficients a j and b j are real-valued functions belonging to the Gevrey class G s (T 1) , s > 1. The global s −hypoellipticity of this system is characterized in terms of Diophantine approximations and the Nirenberg–Treves condition (P). [ABSTRACT FROM AUTHOR]

We obtain new conditions on periodic integrable functions so that their transformed Fourier series belong to L p . This improves the classical Hardy and Bellman results. A counterpart for the Fourier transforms is also established. Our main tool is a new extension of the Hausdorff–Young–Paley inequality for Fourier transforms. [ABSTRACT FROM AUTHOR]

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising measure-consistent multivariate orthonormal polynomials in dependent random variables. Unlike existing PCEs, whether classical or generalized, no tensor-product structure is assumed or required. Important mathematical properties of the generalized PCE are studied by constructing orthogonal decomposition of polynomial spaces, explaining completeness of orthogonal polynomials for prescribed assumptions, exploiting whitening transformation for generating orthonormal polynomial bases, and demonstrating mean-square convergence to the correct limit. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PCE for a general output variable in terms of the expansion coefficients. An example derived from a stochastic boundary-value problem illustrates the generalized PCE approximation in estimating the statistical properties of an output variable for 12 distinct non-product-type probability measures of input variables. [ABSTRACT FROM AUTHOR]

We demonstrate that the Fourier partial sums of a Lipschitz continuous function on the two-dimensional special unitary group converge uniformly to the function. [ABSTRACT FROM AUTHOR]

Previously, the parabolic equation method was used to study the high-frequency acoustic diffraction by a strongly elongated spheroid. This paper represents a continuation of that study. We justify some formal steps of the parabolic equation method at the level typical for the general PDE theory. In particular, we prove that a formal solution of the parabolic equation is actually the classical solution. We prove its uniqueness. We use various asymptotic properties of the higher functions. Some of these properties are new. We study location of zeros of the Whittaker functions. [ABSTRACT FROM AUTHOR]

If g ∈ L 2 ( [ 0 , 2 π ] ) let g ˆ be the sequence of Fourier coefficients of g , let D denote differentiation and let I denote the identity operator. Given α , β ∈ Z , we consider the operator D 2 − i ( α + β ) D − α β I on the second order Sobolev space of L 2 ( [ 0 , 2 π ] ) . The multiplier of this operator is − ( n − α ) ( n − β ) considered as a function of n ∈ Z , so that g ˆ ( α ) = g ˆ ( β ) = 0 for any function g in the range of the operator. Let δ x denote the Dirac measure at x , and let ⁎ denote convolution. If b ∈ [ 0 , 2 π ] let λ b be the measure 2 − 1 [ ( e i b ( α − β 2 ) + e − i b ( α − β 2 ) ] δ 0 − 2 − 1 [ ( e i b ( α + β 2 ) δ b + e − i b ( α + β 2 ) δ − b ] . A function of the form λ b ⁎ f is called a generalised difference , and we let F be the family of functions h such that h is a sum of five generalised differences. It is shown that for g ∈ L 2 ( [ 0 , 2 π ] ) , g ∈ F if and only if g ˆ ( α ) = g ˆ ( β ) = 0 . Consequently, F is a Hilbert subspace of L 2 ( [ 0 , 2 π ] ) and it is the range of D 2 − i ( α + β ) D − α β I . The methods use partitions of intervals and estimates of integrals in Euclidean space. There are applications to the automatic continuity of linear forms in abstract harmonic analysis. [ABSTRACT FROM AUTHOR]

In this paper we improve existing bounds for Siegel modular forms of small weight. [ABSTRACT FROM AUTHOR]

In this study, we introduce the norm-attaining property for a dual pair of Banach spaces. We use this property to determine a quotient Banach space as a dual space. We also apply this property to obtain another proof of the James's characterization theorem regarding the reflexivity of a separable Banach space. In addition, the norm-attaining property of a Fourier space is studied. [ABSTRACT FROM AUTHOR]

An approximation of functions in Müntz spaces M Λ , L 1 of L 1 functions is studied with the help of Fourier series. Müntz spaces are considered with the Müntz condition and the gap condition imposed. It is proved that up to an isomorphism such spaces are contained in Weil–Nagy's class. Moreover, existence of Schauder bases in Müntz spaces M Λ , L 1 is investigated. [ABSTRACT FROM AUTHOR]

We obtain discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type. It is shown that the microlocal properties of an ultradistribution can be obtained by sampling the Fourier transforms of its localizations over a lattice in R d . In particular, we prove the following discrete characterization of the analytic wave front set of a distribution f ∈ D ′ ( Ω ) . Let Λ be a lattice in R d and let U be an open convex neighborhood of the origin such that U ∩ Λ ⁎ = { 0 } . The analytic wave front set W F A ( f ) coincides with the complement in Ω × ( R d ∖ { 0 } ) of the set of points ( x 0 , ξ 0 ) for which there are an open neighborhood V ⊂ Ω ∩ ( x 0 + U ) of x 0 , an open conic neighborhood Γ of ξ 0 , and a bounded sequence ( f p ) p ∈ N in E ′ ( Ω ∩ ( x 0 + U ) ) with f p = f on V such that for some h > 0 sup μ ∈ Γ ∩ Λ ⁡ | f p ˆ ( μ ) | | μ | p ≤ h p + 1 p ! , ∀ p ∈ N . [ABSTRACT FROM AUTHOR]

For functions f in Orlicz classes, we consider multiple Walsh–Fourier series for which the rectangular partial sums S n ( x ; f ) have indices n = ( n 1 , … , n N ) ∈ Z N ( N ≥ 3 ), where either N or N − 1 components are elements of (single) lacunary sequences. For this series, we prove the validity of weak generalized localization almost everywhere on an arbitrary measurable set A ⊂ I N = { x ∈ R N : 0 ≤ x j < 1 , j = 1 , 2 , … , N } , in the case when the structure and geometry of A are defined by the properties B k , 2 ≤ k ≤ N . We define the relation between the parameter k and the “smoothness” of functions in terms of the Orlicz classes. As a consequence, we obtain some results on the “local smoothness conditions.” In particular, the theorem is proved for the convergence of Walsh–Fourier series on an arbitrary open set Ω ⊂ I N under the minimal conditions imposed on the smoothness of the function on this set. [ABSTRACT FROM AUTHOR]

The paper investigates the uniform convergence of negative order Cesaro means of Fourier series of continuous functions after correction of these functions on a set of small measure. We also give an example of continuous function, the ( C , α ) means of Fourier series of which diverge over any subsequence on the set of positive measure for any α ∈ ( − 1 , − 1 2 ) . [ABSTRACT FROM AUTHOR]

A point is out of the support of a distribution on the open standard simplex if and only if the orthogonal expansion of the distribution is uniformly summable in Abel means to zero on a neighborhood of the point. A similar equivalence holds for Fourier–Laplace series on the unit sphere and orthogonal expansions on the open unit ball. [ABSTRACT FROM AUTHOR]

A general summability method of double Fourier series is given with the help of a double sequence θ ( k , n ) . Under some conditions on θ we show that the Marcinkiewicz- θ -means of a function f ∈ L 1 ( T 2 ) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so-called modified Lebesgue points of f ∈ L p ( T 2 ) , whenever 1 < p < ∞ . The sufficient conditions of θ are proved for the Fejér, Abel and Cesàro summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong θ -summability. Some other special cases of the θ -summation are considered as well, such as the Weierstrass, Picard, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations. [ABSTRACT FROM AUTHOR]

We obtain some well-known expansions for the Lerch transcendent and the Hurwitz zeta function using elementary Fourier analytic methods. These Fourier series can be used to analytically continue the functions and prove the classical functional equations, which arise from the relations satisfied by the Fourier conjugate and flat Fourier series. In particular, the functional equation for the Riemann zeta function can be obtained in this way without contour integrals. The conjugate series for special values of the parameters yields analogous results for the Bernoulli and Apostol–Bernoulli polynomials. Finally, we give some consequences derived from the Fourier series. [ABSTRACT FROM AUTHOR]

In the present article we study the rate of convergence problem of Fourier series by Deferred Cesàro Mean in the generalized Hölder metric (Hp(w)) space which was earlier introduced by Das, Nath and Ray. [ABSTRACT FROM AUTHOR]

In this paper we state new theorems of Hardy–Littlewood type for functions with general monotone Fourier coefficients. Sharpness of stated results is discusses. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier–Jacobi expansions in Morrey spaces. [Copyright &y& Elsevier]

Abstract: Let be a sequence of real numbers. Denote the set of points for which the trigonometric series converges absolutely. It is shown that if tends to infinity monotonically, the Hausdorff dimension of is given by the formula ; if not, this dimensional formula may be false. This strengthens a former work of Erdös and Taylor. [Copyright &y& Elsevier]

Abstract: It is shown that if then Lebesgue summability, summability ( ), Abel summability, Riemann summability, and summability ( ) of the series are all equivalent to one another. [Copyright &y& Elsevier]

Abstract: We present an application to random Fourier series for -subgaussian random variables. In particular, we generalize previously known notions of dependence for random variables by introducing the concept of -manageable random variables, and consider Fourier series of -manageable -subgaussian random variables. Gaussian series are then a particular case of the series considered in the paper. Conditions for the uniform convergence of such series in probability are established and a rate of convergence is investigated. Moreover, the results are new even for the case of independent random variables. [Copyright &y& Elsevier]

Abstract: In this paper we prove asymptotic formulas for the norms of and . These products can be expressed using and respectively. We prove an estimate for at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of . [Copyright &y& Elsevier]

Abstract: We derive some 1-D symmetry and uniqueness or non-existence results for nonnegative solutions of in low dimension, under suitable assumptions on and . Our method is based upon a combination of Fourier series and Liouville theorems. [Copyright &y& Elsevier]

Abstract: The main aim of this work is to consider integral equations of convolution type with the Toeplitz plus Hankel kernels firstly posed by Tsitsiklis and Levy (1981) [11]. By constructing eight new generalized convolutions for the finite Hartley transforms we obtain a necessary and sufficient condition for the solvability and unique explicit -solution of those equations. Thanks to this convolution approach the solvability condition obtained here is remarkably different from those in Tsitsiklis and Levy (1981) [11] and in other papers. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the trigonometric Fourier series of two variable integrable functions. That is, let such that (, ) for some and . Then for each integrable function we have the a.e. relation . It will be a straightforward and easy consequence of this result the historical cone restricted a.e. convergence result with respect to the two-dimensional Fejér means of integrable functions due to Marcinkiewicz and Zygmund (1939) . [Copyright &y& Elsevier]

Abstract: Motivated by the notion of Lebesgue summability of trigonometric series, we define the Lebesgue summability of trigonometric integrals in terms of the symmetric differentiability of the sum of the formally integrated trigonometric integral in question. We extend two theorems of Zygmund from trigonometric series to integrals, and one of them even in a more general form. [Copyright &y& Elsevier]

Abstract: We consider the indefinite Sturm–Liouville problem , where satisfies . Conditions are presented such that the (normed) eigenfunctions form a Riesz basis of the Hilbert space (using known results for a modified problem). The main focus is on the non-Riesz basis case: We construct a function having no eigenfunction expansion . Furthermore, a sequence is constructed such that the “Fourier series” does not converge in . These problems are closely related to the regularity property of the closed non-semibounded symmetric sesquilinear form with Dirichlet boundary conditions in where . For the associated operator we construct elements in the difference between and the domain of the associated regular closed form, i.e. . [Copyright &y& Elsevier]

Abstract: Carlos Segovia and Richard Wheeden defined fractional square functions involving fractional derivatives. They obtained characterizations of potential spaces via square functions. Our aim in this paper is to reconsider the ideas of Segovia and Wheeden under the light of the semigroups of operators. We develop a quite general theory of fractional square functions associated to certain classes of operators. We present some examples of differential operators where our theory applies. We recover in a more compact way the results of Segovia and Wheeden and we obtain new characterizations of the potential spaces associated to the harmonic oscillator and Ornstein–Uhlenbeck operators. [Copyright &y& Elsevier]

Abstract: A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from to for all and, consequently, is of weak type , where is depending only on θ and or . As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces and so they converge in norm . Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée–Poussin, Rogosinski and Riesz summations. [Copyright &y& Elsevier]

Abstract: In this paper we study the asymptotic growth behavior of solutions to the Dirac–Hodge equation on upper half-space of . By means of the Fourier transform we introduce lower and upper growth orders and generalizations of the maximum term and central index for this function class. Together with a Cauchy estimate we obtain an explicit lower and upper bound estimate of the maximum modulus in terms of these notions. [Copyright &y& Elsevier]

Abstract: In the present paper we consider the trigonometric series with -general monotone and -rest bounded variation coefficients. Necessary and sufficient conditions of L-convergence for such series are obtained in terms of the coefficients. Moreover, we generalize and extend the Tikhonov results [J. Math. Anal. Appl. 347 (2008) 416–427] to the class or the class . [ABSTRACT FROM AUTHOR]

Abstract: Motivated by the recent work on the non-harmonic Fourier atoms initiated by T. Qian and the non-harmonic Fourier series which originated from the celebrated work of Paley and Wiener, we introduce an integral version of the non-harmonic Fourier series, called Chirp transform. As an integral transform with kernel , the Chirp transform is an unitary isometry from onto and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take as a basis which in some sense is dual to the theory of non-harmonic Fourier series which take as a basis. The Chirp version of the Shannon sampling theorem and the Poisson summation formula are also considered by dealing with sampling points which may non-equally distributed. Since the Chirp transform interchanges weighted derivatives into multiplications, it plays a role in solving certain differential equations with variable coefficients. In addition, we extend T. Qian''s theorem on the characterization of a measure to be a linear combination of a number of harmonic measures on the unit disc with positive integer coefficients to that with positive rational coefficients. [Copyright &y& Elsevier]

Abstract: In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions. [ABSTRACT FROM AUTHOR]

Abstract: Firstly, we study the uniform convergence of cosine and sine Fourier transforms. Secondly, we obtain Pitt–Boas type results on -integrability of Fourier transforms with the power weights. The solutions of both problems are written as criteria in terms of general monotone functions. [Copyright &y& Elsevier]

Abstract: In this paper we investigate almost-everywhere convergence properties of the Bochner–Riesz means of N-fold Fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials. It is proved that if the order of the Bochner–Riesz means , then the Bochner–Riesz means of a function , converge to zero almost-everywhere on . [ABSTRACT FROM AUTHOR]

Abstract: We study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials. The considered dissipation is given by the sum of the memory term and the damping term. When the dissipation is effective over the whole system, we show that the solution decays in at the rate as , provided that the corresponding initial data are in , where n is the space dimension. The proof is based on the energy method in the Fourier space. Also, we discuss similar systems with weaker dissipation by introducing the operator with in front of the dissipation terms and observe that the decay structure of these systems is of the regularity-loss type. [Copyright &y& Elsevier]

Abstract: We extend the results of A.S. Belov from single to double Fourier series, which give necessary conditions in terms of the Fourier coefficients for -convergence. Our basic tools are Hardy''s inequality for the Taylor coefficients of a function in the Hardy space on the unit disk, and the Bernstein–Zygmund inequalities for the derivative of a trigonometric polynomial in -norm. [Copyright &y& Elsevier]

Abstract: This article considers the dynamic equation of a reduced model for thin-film micromagnetics deduced by A. DeSimone, R.V. Kohn and F. Otto in [A. DeSimone, R.V. Kohn, F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1–53]. To derive the existence of weak solutions under periodical boundary condition, the authors first prove the existence of smooth solutions for the approximating equation, then prove the convergence of the viscosity solution when the viscosity term vanishes, which implies the existence of solutions for the original equation. [Copyright &y& Elsevier]

Abstract: A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as for some , then the series cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series with as . [Copyright &y& Elsevier]

Abstract: For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its application. [Copyright &y& Elsevier]

Abstract: In this paper we study the general localization principle for Fourier–Laplace series on unit sphere . Weak type property of maximal functions is used to establish the estimates of the maximal operators of Riesz means at critical index . The properties Jacobi polynomials are used in estimating the maximal operators of spectral expansions in . For extending positive results on critical line , , we apply interpolation theorem for the family of the linear operators of weak types. The generalized localization principle is established by the analysis of spectral expansions in . We have proved the sufficient conditions for the almost everywhere convergence of Fourier–Laplace series by Riesz means on the critical line. [Copyright &y& Elsevier]

Abstract: Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia''s extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues. [Copyright &y& Elsevier]

Abstract: Harmonic analysis has been an efficient tool in control theory for a long time, see, e.g., Russell [D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev. 20 (1978) 639–739] and its numerous references. Here we establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened gap condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams. [Copyright &y& Elsevier]

Abstract: In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303–308], which states that if K is a convex domain in the plane with perimeter and area , then one gets , where denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc. [Copyright &y& Elsevier]

Abstract: We consider high stresses in stiff-fiber reinforced materials, which increase rapidly as fibers approximate to one another. This paper presents the optimal blow-up rate of the stresses with respect to the distance between a pair of stiff fibers in . The blow-up result plays an important role in our understanding of low strengths of fiber-reinforced composites. Referring to a problem of anti-plane shear, the stresses can be interpreted as the electric fields outside closely spaced perfect conductors in , under the action of applied electric field ∇H. It has been shown by Ammari, Kang et al. that in the particular case of circular inclusions, the electric field blows up at the optimal rate as , where ϵ is the distance between conductors. Recently, Yun has extended the blow-up result to pairs of conductors associated with a large class of shapes whose complements can be transformed conformally to the outside of a circle with mapping. However, it presented a suboptimal result that only for a special uniform field , the electric fields blow up at the exact rate . In this paper, an upper bound with the rate of electric field for any harmonic function H is established. This yields the optimal blow-up rate for the inclusions in the same class of shapes as Yun. [Copyright &y& Elsevier]