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Closure of dilates of shift-invariant subspaces

Moisés Soto-Bajo (2013)

Open Mathematics

Let V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a...

Commutators based on the Calderón reproducing formula

Krzysztof Nowak (1993)

Studia Mathematica

We prove the Schatten-Lorentz ideal criteria for commutators of multiplications and projections based on the Calderón reproducing formula and the decomposition theorem for the space of symbols corresponding to commutators in the Schatten ideal.

Compactly supported frames for spaces of distributions associated with nonnegative self-adjoint operators

S. Dekel, G. Kerkyacharian, G. Kyriazis, P. Petrushev (2014)

Studia Mathematica

A small perturbation method is developed and employed to construct frames with compactly supported elements of small shrinking support for Besov and Triebel-Lizorkin spaces in the general setting of a doubling metric measure space in the presence of a nonnegative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. This allows one, in particular, to construct compactly supported frames for Besov and Triebel-Lizorkin spaces on the sphere, on the interval with...

Completeness in L1(R) of discrete translates.

Joaquim Bruna, Alexander Olevskii, Alexander Ulanovskii (2006)

Revista Matemática Iberoamericana

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a function φ ∈ L1(R) such that its Λ translates φ(x - λ), λ ∈ Λ, span L1(R). It is shown that these spectra coincide with the uniqueness sets for certain analytic clases. We also present examples of discrete spectra Λ ∈ R which do not admit a single generator while they admit a pair of generators.

Compression of satellite data.

Roberto Barrio, Antonio Elipe (2002)

Revista Matemática Complutense

In this paper, we present the simple and double compression algorithms with an error control for compressing satellite data corresponding to several revolutions. The compressions are performed by means of approximations in the norm L∞ by finite series of Chebyshev polynomials, with their known properties of fast evaluation, uniform distribution of the error, and validity over large intervals of time. By using the error control here introduced, the number of terms of the series is given automatically...

Conjugacy for Fourier-Bessel expansions

Óscar Ciaurri, Krzysztof Stempak (2006)

Studia Mathematica

We define and investigate the conjugate operator for Fourier-Bessel expansions. Weighted norm and weak type (1,1) inequalities are proved for this operator by using a local version of the Calderón-Zygmund theory, with weights in most cases more general than A p weights. Also results on Poisson and conjugate Poisson integrals are furnished for the expansions considered. Finally, an alternative conjugate operator is discussed.

Conjugate martingale transforms

Ferenc Weisz (1992)

Studia Mathematica

Characterizations of H₁, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

Hans Weber (2007)

Open Mathematics

Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas....

Connectivity, homotopy degree, and other properties of α-localized wavelets on R.

Gustavo Garrigós (1999)

Publicacions Matemàtiques

In this paper, we study general properties of α-localized wavelets and multiresolution analyses, when 1/2 < α ≤ ∞. Related to the latter, we improve a well-known result of A. Cohen by showing that the correspondence m → φ' = Π1∞ m(2−j ·), between low-pass filters in Hα(T) and Fourier transforms of α-localized scaling functions (in Hα(R)), is actually a homeomorphism of topological spaces. We also show that the space of such filters can be regarded as a connected infinite dimensional manifold,...

Consistency of trigonometric and polynomial regression estimators

Waldemar Popiński (1998)

Applicationes Mathematicae

The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials e k , k=0,1,..., for the observation model y i = f ( x i ) + η i , i=1,...,n, where the η i are independent random variables with zero mean value and finite variance, and the observation points x i ∈ [ a , b ] , i=1,...,n, form a random sample from a distribution with density ϱ ∈ L 1 [ a , b ] . Sufficient and necessary conditions are obtained for consistency in the sense of the errors ∥ f - f ^ N ∥ , | f ( x ) - N ( x ) | , x ∈ [ a , b ] ,...

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