Reiteration Formulae for the Real Interpolation Method Including L or R Limiting Spaces (original) (raw)
Approximation by quasi-interpolation operators and Smolyak's algorithm
Journal of Complexity, 2021
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 Lq-norm for functions from the Besov spaces B s p,θ (T d) and the Triebel-Lizorkin spaces F s p,θ (T d) for all s > 0 and admissible 1 ≤ p, θ ≤ ∞ as well as provide analogues of the Littlewood-Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.
Proceedings of the American Mathematical Society, Series B
In this paper we consider interpolation in model spaces, H 2 BH 2 with B a Blaschke product. We study unions of interpolating sequences for two sequences that are far from each other in the pseudohyperbolic metric as well as two sequences that are close to each other in the pseudohyperbolic metric. The paper concludes with a discussion of the behavior of Frostman sequences under perturbations.
Modes of Convergence: Interpolation Methods I
Journal of Approximation Theory, 2001
In the present paper we explore an approximation theoretic approach to some classical convergence theorems of real analysis. The background of this paper is the intuition that some of the usual compactness theorems on various modes of convergence in classical analysis are based on suitable ways of obtaining good decompositions of functions to exploit rates of approximation, cancellations, or appropriate control of sizes that can be controlled by the basic functionals of real interpolation.
Non-linear Approximation and Interpolation Spaces
Journal of Approximation Theory, 2001
We study n-term wavelet-type approximations in Besov and Triebel-Lizorkin spaces. In particular, we characterize spaces of functions which have prescribed degree of n-term approximation in terms of interpolation spaces. These results are applied to identify interpolation spaces between Triebel-Lizorkin and Besov spaces.
Limit problems for interpolation by analytic radial basis functions
Journal of Computational and Applied Mathematics, 2008
Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee,Yoon, andYoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite-Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.
On a convergence of the Fourier-Pade interpolation
We investigate convergence of the rational-trigonometric-polynomial interpolation that performs convergence acceleration of the classical trigonometric interpolation by sequential application of polynomial and rational correction functions. Unknown parameters of the rational corrections are determined along the ideas of the Fourier-Pade approximations. The resultant interpolation we call as Fourier-Pade interpolation and investigate its convergence in the regions away from the endpoints. Comparison with other rational-trigonometricpolynomial interpolations outlines the convergence properties of the Fourier-Pade interpolation.
Real Interpolation of Vector-Valued Spaces in Non-Diagonal Case
Proceedings of the American Mathematical Society, 2005
It is shown that the formula (l s 0 p 0 (A 0), ..., l sn pn (An)) θ,q = l s q ((A 0 , ..., An) θ,q), where θ = (θ 0 , ..., θn) and s = θ 0 s 0 + ... + θnsn is correct under the restrictions A n−1 = An and s n−1 = sn. It is also true if we suppose that
Spaces Λα(X) and interpolation
Journal of Functional Analysis, 1972
For two pairs of rearrangement invariant spaces D = [(X, , Yr), (X, , Ya)] we give necessary and sufficient conditions for pairs (X, Y) to be weak intermediate for o, i.e., each operator which is of weak types (Xi, YJ, i = 1, 2, also maps X boundedly to Y. Spaces A,(X) are introduced and are shown to have many of the properties that characterize Lorentz L*'r spaces. Necessary and sufficient conditions in terms of a simple function F(s, t) are given in order that (d,(X), A,(Y)) be weak intermediate for o. Other properties of the function F(s, t) yield sufficient conditions and necessary conditions for interpolation theorems.
Interpolation of Operators for a Spaces
2000
Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S {r) dr,-" , VaC'Wo Y a (r) dr.
Interpolation In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate (i.e., estimate) the value of that function for an intermediate value of the independent variable. A different problem which is closely related to in terpolation is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complex to evaluate efficiently. A few known data points from the original function can be used to create an interpolation based on a simpler function. Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in precision. In the examples below if we consider as a topological space and the function forms a different kind of Banach spaces then the problem is treated as "interpolation of operators". The classical results about interpolation of operators are the Riesz–Thorin theorem and the