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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]

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: We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund''s monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series. [Copyright &y& Elsevier]

Abstract: In this paper, using techniques of Nevanlinna theory, we give a degeneracy theorem of meromorphic mappings of into with few hyperplanes and truncated multiplicities. [Copyright &y& Elsevier]

Abstract: We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin. [Copyright &y& Elsevier]