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Necessary conditions for the L p -convergence ( 0 < p < 1 ) of single and double trigonometric series

Xhevat Z. Krasniqi, Péter Kórus, Ferenc Móricz (2014)

Mathematica Bohemica

We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the L p -metric, where 0 < p < 1 . The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in H p and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the L p -metric, where 0 < p < 1 .

Notes on interpolation of Hardy spaces

Quanhua Xu (1992)

Annales de l'institut Fourier

Let H p denote the usual Hardy space of analytic functions on the unit disc ( 0 &lt; p ) . We prove that for every function f H 1 there exists a linear operator T defined on L 1 ( T ) which is simultaneously bounded from L 1 ( T ) to H 1 and from L ( T ) to H such that T ( f ) = f . Consequently, we get the following results ( 1 p 0 , p 1 ) :1) ( H p 0 , H p 1 ) is a Calderon-Mitjagin couple;2) for any interpolation functor F , we have F ( H p 0 , H p 1 ) = H ( F ( L p 0 ( T ) , L p 1 ( T ) ) ) , where H ( F ( L p 0 ( T ) , L p 1 ( T ) ) ) denotes the closed subspace of F ( L p 0 ( T ) , L p 1 ( T ) ) of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy...

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