We study approximation of multivariate periodic functions from Besov and Triebel--Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the $L_q$-norm for functions from the Besov spaces $\mathbf{B}_{p,θ}^s(\mathbb{T}^d)$ and the Triebel--Lizorkin spaces $\mathbf{F}_{p,θ}^s(\mathbb{T}^d)$ for all $s>0$ and admissible $1\le p,θ\le \infty$ as well as provide analogues of the Littlewood--Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.
We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetildeφ_j$ and trigonometric polynomials $φ_j$. The class of such operators includes classical interpolation polynomials ($\widetildeφ_j$ is the Dirac delta function), Kantorovich-type operators ($\widetildeφ_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on $\widetildeφ_j$ and $φ_j$, we obtain upper and lower bound estimates for the $L_p$-error of a…
In this paper, we prove that the inner complex interpolation of two quasi-Banach lattices coincides with the closure of their intersection in their Calderón product. This generalizes a classical result by Shestakov in 1974 for Banach lattices.
We consider the Calderón-Lozanovskii construction $φ(X_0,X_1)$ in the context of quasi-Banach lattices and provide an extension of a result by V. I. Ovchinnikov concerning the associated interpolation methods $φ^c$ and $φ^0$. Our approach is based on the interpolation properties of $(\infty,1)$-regular operators between quasi-Banach lattices.
We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and project scheme that produces it. On the other hand, if an LCA group G admits a simple quasicrystal, we prove that recent results of Meyer and Matei for the case of the n-dimensional Euclidean space can be extended to G. More precisely, we prove that simple quasicrystals are universal sets of stable sampling and universal sets of stable interpolation in genera…
The structure of certain types of quasi shift-invariant spaces, which take the form $V(ψ,\mathcal{X}):=\overline{\text{span}}^{L_2}\{ψ(\cdot-x_j):j\in\mathbb{Z}\}$ for a discrete set $\mathcal{X}=(x_j)\subset\mathbb{R}$ is investigated. Additionally, the relation is explored between pairs $(ψ,\mathcal{X})$ and $(ϕ,\mathcal{Y})$ such that interpolation of functions in $V(ψ,\mathcal{X})$ via interpolants in $V(ϕ,\mathcal{Y})$ solely from the samples of the original function is possible and stable. Some conditions are given for which the sampling problem is stable, and for which recovery of funct…
In this article, the authors study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e.\,g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the $\pm$-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasize is given to the quasi-Banach case and to the interpolation property.
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ as well as it is bounded on both the weak ball quasi-Banach function space $WX$ and the associated space, the authors establish various Littlewood--Paley function characterizations of $WH_X({\mathbb R}^n)$ under some weak assumptions on the Littlewood--Paley functions. The authors also prove that the real interpolation intermediate space $(H_{X}({\mathbb R}^n),L^\infty({\mathbb R}^…
In this paper we use the values of the Riemann $Z(t)$-function in order to construct certain quasi-orthonormal system of vectors. On this basis we prove a formula for microscopic interpolation of the function $Z(t)$. Simultaneously we have obtained a new algorithm how to construct asymptotic solutions of Diophantine equation with Leibnitz coefficients. This paper is English version of the paper \cite{3} except the Appendix.
In this paper, we introduce a new scale of tent spaces which covers, the (weighted) tent spaces of Coifman-Meyer-Stein and of Hofmann-Mayboroda-McIntosh, and some other tent spaces considered by Dahlberg, Kenig-Pipher and Auscher-Axelsson in elliptic equations. The strong factorizations within our tent spaces, with applications to quasi-Banach complex interpolation and to multiplier-duality theory, are established. This way, we unify and extend the corresponding results obtained by Coifman-Meyer-Stein, Cohn-Verbitsky and Hytönen-Rosén.
Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…
Let $\vec{X}=(X_0, X_1)$ be a compatible couple of Banach spaces, $ 1\le p \le \infty$ and let $ φ$ be positive quasi-concave function. Denote by $\overline{X}_{φ,p}=(X_0,X_1)_{φ,p}$ the real interpolation spaces defined by S. Janson (1981). We give necessary and sufficient conditions on $ φ_{0}$, $φ_{1}$ and $φ$ for the validity of \begin{equation*} \left(\overline{X}_{φ_{0},1},\overline{X}_{φ_{1},1} \right) _{φ,p}= \left(\overline{X}_{φ_{0},\infty},\overline{X}_{φ_{1},\infty}\right)_{φ,p} \end{equation*} for all $ 1\le p\le \infty$, and all Banach couples $\overline{X}. $
An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, $S(X,μ)$, of all $μ$-measurable simple functions on $σ$- finite measure space $(X,μ)$ into $M(Y,ν)$, the class of $ν$-measurable functions on $σ$- finite measure space $(Y,ν)$, and satisfies endpoint estimates of type: $1 < p< \infty$, $1 \leq r < \infty$, \begin{equation*} λ\, ν\left( \left\lbrace y \in Y : |(Tf)(y)| > λ\right\rbrace \right)^{\frac{1}{p}} \leq C_{p,r}…
An ensemble of quasi-periodic discrete Schrödinger operators with an arbitrary number of basic frequencies is considered, in a lattice of arbitrary dimension, in which the hull function is a realisation of a stationary Gaussian process on the torus. We show that, for almost every element of the ensemble, the quasi-periodic operator boasts Anderson localization with simple pure point spectrum at strong coupling. One of the ingredients of the proof is a new lower bound on the interpolation error for stationary Gaussian processes on the torus (also known as local non-determinism).
Let $Λ\subset R$ be a strictly increasing sequence. For $r = 1,2$, we give a simple explicit expression for an equivalent norm on the trace spaces $W_p^r(R)|_Λ$, $L_p^r(R)|_Λ$ of the non-homogeneous and homogeneous Sobolev spaces with $r$ derivatives $W_p^r(R)$, $L_p^r(R)$. We also construct an interpolating spline of low degree having optimal norm up to a constant factor. A general result relating interpolation in $L^r_p(R)$ and $W^r_p(R)$ for all $r \geq 1$ is also given.
Let $ϕ$ be a quasiconformal mapping, and let $T_ϕ$ be the composition operator which maps $f$ to $f\circϕ$. Since $ϕ$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_ϕ$ on $L^p$ and $W^{1,p}$ for $1<p<\infty$. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in $H^{s,p}$ are sent to $H^{s,q}$ whenever $0<s<1$ for appropriate values of $q$. The techniques used lead to sharp res…
