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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]

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]

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]

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]

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: 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: 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: 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: 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 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: The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general -spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of positive doubly periodic solutions for nonlinear telegraph system is discussed using the method of upper and lower solutions. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces , . We consider the classical coefficient conditions, the Marcinkiewicz–Hörmander–Mihlin conditions. They are known to be sufficient for the trigonometric system in the one and two-dimensional cases for the spaces , . This can be found in the original papers of Marcinkiewicz [J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939) 78–91], Hörmander [L. Hörmander, Estimates for translation invariant operators in spaces, Acta Math. 104 (1960) 93–140], and Mihlin [S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956) 701–703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces. [Copyright &y& Elsevier]

Abstract: In this paper we consider the trigonometric Fourier series with the β-general monotone coefficients. Necessary and sufficient conditions of -convergence for such a series, that is , are obtained in terms of coefficients. [Copyright &y& Elsevier]

Abstract: Let be the Littlewood–Paley g-function of f on . In this paper, the authors prove that if (the space of functions with bounded mean oscillation), then is either infinite everywhere or finite almost everywhere, and in the latter case, is bounded from into (the space of functions with bounded lower oscillation), which is a proper subspace of . Moreover, the authors also establish similar results for the Lusin-area function and the Littlewood–Paley -function. [Copyright &y& Elsevier]

Abstract: We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs to one of the Zygmund classes and for some . This paper is a natural supplement to our earlier one [F. Móricz, Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl. 324 (2) (2006) 1168–1177] under the same title, and we keep its notations. [Copyright &y& Elsevier]

Abstract: In this paper, we generalize some well-known results (Theorems A, C, and D) by establishing two general results (Theorems 1 and 3). As special applications, we find that the (generalized) jumps of f can be determined by the higher order partial derivatives of its Abel–Poisson means. This is different from the determination of jumps by higher order derivatives of the partial sums. We also give some estimates of the higher order partial derivatives of the Abel–Poisson mean of an integrable function F at those points at which F is smooth. [Copyright &y& Elsevier]

Abstract: In this paper we study Clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure, or more generally, at least a pin structure. The examples treated here are manifolds that can be parametrized by where U is a subdomain of either or and Γ is a Kleinian group acting discontinuously on U. The examples studied here include and the Hopf manifolds . Also some hyperbolic manifolds will be treated. Special kinds of Clifford-analytic automorphic forms associated to the different choices of Γ are used to construct explicit Cauchy kernels, Cauchy integral formulas, Green''s kernels and formulas together with Hardy spaces and Plemelj projection operators for spaces of hypersurfaces lying in these manifolds. [Copyright &y& Elsevier]

Abstract: In this paper, we examine a class of averaging operators which exhibit local smoothing. That is, viewed as a function of space and time variables, the operators yield more smoothing than the fixed-time estimates. Sogge showed in a more general setting that if these operators satisfy a cinematic curvature condition, they will exhibit some local smoothing [C.D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991) 231–251]. Here we translate this condition into the setting of averaging operators in the plane. We prove that cinematic curvature is not necessary for local smoothing to occur, exhibiting a class of operators which fail the cinematic curvature condition but still satisfy a local smoothing estimate. Furthermore, the amount of local smoothing exhibited by these operators is strictly less than that conjectured for operators satisfying the cinematic curvature condition. [Copyright &y& Elsevier]

Abstract: We generalize a Hardy–Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions. [Copyright &y& Elsevier]