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Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1

Leonardo Colzani, Peter Sjögren (1999)

Studia Mathematica

We study convolution operators bounded on the non-normable Lorentz spaces L 1 , q of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on L 1 , q . In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...

Transplantation operators and Cesàro operators for the Hankel transform

Yuichi Kanjin (2006)

Studia Mathematica

The transplantation operators for the Hankel transform are considered. We prove that the transplantation operator maps an integrable function under certain conditions to an integrable function. As an application, we obtain the L¹-boundedness and H¹-boundedness of Cesàro operators for the Hankel transform.

Trigonometric approximation by Nörlund type means in L p -norm

Bogdan Szal (2009)

Commentationes Mathematicae Universitatis Carolinae

We show that the same degree of approximation as in the theorems proved by L. Leindler [Trigonometric approximation in L p -norm, J. Math. Anal. Appl. 302 (2005), 129–136] and P. Chandra [Trigonometric approximation of functions in L p -norm, J. Math. Anal. Appl. 275 (2002), 13–26] is valid for a more general class of lower triangular matrices. We also prove that these theorems are true under weakened assumptions.

Two complete and minimal systems associated with the zeros of the Riemann zeta function

Jean-François Burnol (2004)

Journal de Théorie des Nombres de Bordeaux

We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the “dual Poisson formula” of Duffin-Weinberger (also named by us co-Poisson formula), and the “Sonine spaces” of entire functions defined and studied by de Branges. We determine in which (extended) Sonine spaces the zeros define a complete, or minimal, system. We obtain some general results dealing with the distribution of the zeros of the de-Branges-Sonine entire functions. We...

Two examples of subspaces in L 2 l spanned by characters of finite order

Mats Erik Andersson (2001)

Colloquium Mathematicae

By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into L 2 l of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

Two problems related to the non-vanishing of L ( 1 , χ )

Paolo Codecà, Roberto Dvornicich, Umberto Zannier (1998)

Journal de théorie des nombres de Bordeaux

We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or ( ( x y / q ) ) , ( 0 ≤ x , y &lt; q ) , where ( ( u ) ) = u - [ u ] - 1 / 2 denotes the “centered” fractional part of x . These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet L -functions at s = 1 .

Two random constructions inside lacunary sets

Stefan Neuwirth (1999)

Annales de l'institut Fourier

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is Λ ( p ) for all p but is not a Rosenthal set. This holds also for the sequence of primes.

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