This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam s...

We study Besov spaces 5°(LP(Í2)), 0<p,q,a<oo,on domains Í2 in Rd . We show that there is an extension operator W which is a bounded mapping from B°(LP(Ü.)) onto B%(Lp(Rd)). This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of B%(Lp(il)), atomic decompositions for the elements of 5°(Lp(f2)), and a description of the Besov spaces by means...

We study the construction of extrapolation spaces operator-valued multiplication operators on Bochner Lp-spaces by means Lp-fiber spaces.

We provide lower bounds for the norms of embeddings between γ-weighted Anchored and ANOVA spaces of s-variate functions with mixed partial derivatives of order one bounded in Lp norm (p ∈ [1,∞]). In particular we show that the norms behave polynomially in s for Finite Order Weights and Finite Diameter Weights if p > 1, and increase faster than any polynomial in s for Product Order-Dependent Wei...

In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces Lp((0, 1), x2ν+1 dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce th...

It is well known that (see [9]) Cesaro means of 2π-periodic functions f ∈ Lp(T) (1 ≤ p ≤ ∞) converges by norms. Hereby T is denoted the interval (−π,π). The problem of the mean summability in weighted Lebesgue spaces has been investigated in [6]. A 2π-periodic nonnegative integrable function w : T→R1 is called a weight function. In the sequel by L p w(T), we denote the Banach function space of ...

Generalized translation operators for orthogonal expansions with respect to families of weight functions on the unit ball and on the standard simplex are studied. They are used to define convolution structures and modulus of smoothness for these regions, which are in turn used to characterize the best approximation by polynomials in the weighted Lp spaces. In one variable, this becomes the gene...

The idea of replacing functions by linear operators, the process of quantization, goes back to the foundations of quantum mechanics and has a great impact in mathematics. This applies for instance to representation theory, operator algebra, noncommutative geometry, quantum and free probability or operator space theory. The quantization of measure theory leads to the theory of Lp spaces defined ...

Let Ω⊂R2 be a bounded, simply connected domain with boundary ∂Ω of class C 1,1 except at finitely many points S j where ∂Ω is locally a corner of aperture α j ≤ π2 . Improving on results of Grisvard [13, 14], we show that the solution GΩ f to the Dirichlet problem on Ω with data f ∈ Lp(Ω) and homogeneous boundary conditions satisfies the estimates ‖GΩ f ‖W2,p(Ω) ≤ Cp‖ f ‖Lp(Ω), ∀ 2≤ p <∞, ‖DGΩ ...

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an Lp-space, then it is either a Lp-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non–Hilbertian complemented operator subspaces of non commutative Lp-spaces and show that this class is much richer than in the commutative case. We investigat...