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Quantized orthonormal systems: A non-commutative Kwapień theorem

J. García-Cuerva, J. Parcet (2003)

Studia Mathematica

The concepts of Riesz type and cotype of a given Banach space are extended to a non-commutative setting. First, the Banach space is replaced by an operator space. The notion of quantized orthonormal system, which plays the role of an orthonormal system in the classical setting, is then defined. The Fourier type and cotype of an operator space with respect to a non-commutative compact group fit in this context. Also, the quantized analogs of Rademacher and Gaussian systems are treated. All this is...

Random perturbations of exponential Riesz bases in L 2 ( - π , π )

Gennadii Chistyakov, Yura Lyubarskii (1997)

Annales de l'institut Fourier

Let a sequence { λ n } ⊂ ℝ be given such that the exponential system { exp ( i λ n x ) } forms a Riesz basis in L 2 ( - π , π ) and { ξ n } be a sequence of independent real-valued random variables. We study the properties of the system { exp ( i ( λ n + ξ n ) x ) } as well as related problems on estimation of entire functions with random zeroes and also problems on reconstruction of bandlimited signals with bandwidth 2 π via their samples at the random points { λ n + ξ n } .

Recovery of band-limited functions on locally compact Abelian groups from irregular samples

H. G. Feichtinger, S. S. Pandey (2003)

Czechoslovak Mathematical Journal

Using the techniques of approximation and factorization of convolution operators we study the problem of irregular sampling of band-limited functions on a locally compact Abelian group G . The results of this paper relate to earlier work by Feichtinger and Gröchenig in a similar way as Kluvánek’s work published in 1969 relates to the classical Shannon Sampling Theorem. Generally speaking we claim that reconstruction is possible as long as there is sufficient high sampling density. Moreover, the iterative...

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae

Let P k be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients a k in f = ∑ k a k P k . A systematic use of the basic properties (including some nonstandard ones) of the polynomials P k results in obtaining a low order of the recurrence.

Riesz sequences and arithmetic progressions

Itay Londner, Alexander Olevskiĭ (2014)

Studia Mathematica

Given a set of positive measure on the circle and a set Λ of integers, one can ask whether E ( Λ ) : = e λ ∈ Λ i λ t is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step ℓ = O ( N 1 - ε ) with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain...

Rotation invariant subspaces of Besov and Triebel-Lizorkin space: compactness of embeddings, smoothness and decay of functions.

Leszek Skrzypczak (2002)

Revista Matemática Iberoamericana

Let H be a closed subgroup of the group of rotation of Rn. The subspaces of distributions of Besov-Lizorkin-Triebel type invariant with respect to natural action of H are investigated. We give sufficient and necessary conditions for the compactness of the Sobolev-type embeddings. It is also proved that H-invariance of function implies its decay properties at infinity as well as the better local smoothness. This extends the classical Strauss lemma. The main tool in our investigations is an adapted...

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