On the Lagrange interpolation for a subset ofC k functions

We study the optimal order of approximation forC ^k piecewise analytic functions (cf. Definition 1.2) by Lagrange interpolation associated with the Chebyshev extremal points. It is proved that the Jackson order of approximation is attained, and moreover, ifx is away from the singular points, the local order of approximation atx can be improved byO(n ^?1). Such improvement of the local order of approximation is also shown to be sharp. These results extend earlier results of Mastroianni and Szabados on the order of approximation for continuous piecewise polynomial functions (splines) by the Lagrange interpolation, and thus solve a problem of theirs (about the order of approximation for |x|³) in a much more general form.     

Application of wavelet bases in linear and nonlinear approximation to functions from Besov spaces

Linear and nonlinear approximations to functions from Besov spaces B _{p, q} ^σ ([0, 1]), σ > 0, 1 ≤ p, q ≤ ∞ in a wavelet basis are considered. It is shown that an optimal linear approximation by a D-dimensional subspace of basis wavelet functions has an error of order D ^{-min(σ, σ + 1/2 ? 1/p)} for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). An original scheme is proposed for optimal nonlinear approximation. It is shown how a D-dimensional subspace of basis wavelet functions is to be chosen depending on the approximated function so that the error is on the order of D ^?σ for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). The nonlinear approximation scheme proposed does not require any a priori information on the approximated function.     

Quasi-interpolation by thin-plate splines on a square

Quasi-interpolation is one method of generating approximations from a space of translates of dilates of a single function ψ. This method has been applied widely to approximation by radial basis functions. However, such analysis has most often been performed in the setting of an infinite uniform grid of centers. In this paper we develop general error bounds for approximation by quasiinterpolation on ann-cube. The quasi-interpolant analyzed involves a finite number, growing ash ^?n , of translates of dilates of the function ψ, and a bounded number of edge functions. The centers of the translates of dilates of ψ form a uniformly spaced grid within the cube. These error bounds are then applied to approximation by thin-plate splines on a square. The result is an O(ω(f, [-1,1]²,h)) error bound for approximation by thin-plate splines supplemented with eight arctan functions.     

On the approximation of curves with polylines

The exact values of the estimation of the approximation error of parametrically defined curves by inscribed polylines in them-dimensional space R ^m for classes of functions defined by moduli of continuity are presented. The result is a sort of generalization of the results of B.N. Malozemov on the approximation of continuous functions with polylines. Also, the problem of finding the upper bounds of deviations of parametrically defined curves for this class is solved based on the assumption that these curves intersect at N (N ≥ 2) points of the partition of [0, L]. In the case of m = 2, from the obtained results follow the previous results on the approximation of plane curves with polylines in Euclidean, Hausdorff, and Hamming metrices.     

Optimal approximation of convex curves by functions which are piecewise linear

In this paper an efficient method is presented for solving the problem of approximation of convex curves by functions that are piecewise linear, in such a manner that the maximum absolute value of the approximation error is minimized. The method requires the curves to be convex on the approximation interval only. The boundary values of the approximation function can be either free or specified. The method is based on the property of the optimal solution to be such that each linear segment approximates the curve on its interval optimally while the optimal error is uniformly distributed among the linear segments of the approximation function. Using this method the optimal solution can be determined analytically to the full extent in certain cases, as it was done for functions x² and $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. In general, the optimal solution has to be computed numerically following the procedure suggested in the paper. Using this procedure, optimal solutions were computed for functions sin x, tg x, and arc tg x. Optimal solutions to these functions were used in practical applications.     

An optimal sequential algorithm for the uniform approximation of convex functions on [0,1]2

In this paper an algorithm is given for the sequential selection ofN nodes (i.e., measurement points) for the uniform approximation (recovery) of convex functions over [0, 1]², which has almost optimal order global error, (≦c ₁ N ^?1 lgN), over a naturally defined class of convex functions. This shows the essential superiority of sequential algorithms for this class of approximation problems because any simultaneous choice ofN nodes leads to a global error >c ₀ N ^?1/2. New construction and estimation methods are presented, with possible (e.g., multidimensional) generalizations.     

Directional K- Functionals Claudia Cottin

When considering approximation of continuous periodic functions f: R ^d → R by blending-type approximants which depend on directions ξ₁,…,ξ_ν ∈ R ^d directional moduli of smoothness (1) are appropriate measures of smoothness of /. In this paper, we introduce equivalent directional K- functionals. As an application, we obtain a result on the degree of approximation by certain trigonometric blending functions.     

Un algorithme probabiliste de calcul d'approximations polynômiales sur un hypercube

We describe a Monte Carlo method which enables an iterative computation of the L² approximation of a function on any orthonormal basis. We use it for the approximation of smooth functions on an hypercube with the help of multidimensional orthogonal polynomial basis containing only few terms. The algorithm is both a tool for approximation and numerical integration. To cite this article: S. Maire, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A decomposition into atoms of distributions on spaces of homogeneous type

A maximal function is introduced for distributions acting on certain spaces of Lipschitz functions defined on spaces of homogeneous type. A decomposition into atoms for distributions whose maximal functions belong to L^p, p ? 1, is obtained, as well as, an approximation theorem of these distributions by Lipschitz functions.     

Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators Studied Asymptotically

An Nth order asymptotic expansion is established for the error of weak approximation of a special class of functions by the well-known Cardaliaguet-Euvrard neural network operators. This class is made out of functions f that are N times continuously differentiable over R, so that all f,f′,…, f ^(N) have the same compact support and f ^(N) is of bounded variation. This asymptotic expansion involves products of integrals of the network activation bell-shaped function b and f. The rate of the above convergence depends only on the first derivative of involved functions.     

Adaptive wavelet methods for the stochastic Poisson equation

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ? ^d . The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.     

Hardy approximation toL ∞ functions on subsets of the circle

We consider approximation ofL ^∞ functions byH ^∞ functions on proper substs of the circle. We derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset.     

Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X^∗ of a Banach space X if X^∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X^∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X^∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X^∗.     

Plurifinely Plurisubharmonic Functions and the Monge Ampère Operator

We will define the Monge-Ampère operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets U???? ⁿ and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that in the Dirichlet norm weakly plurifinely plurisubharmonic functions are locally limits of plurisubharmonic functions. As a consequence of the latter, weakly plurifinely plurisubharmonic functions are strongly plurifinely plurisubharmonic outside of a pluripolar set.     

Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

An example of a complete orthonormal system in a Hilbert space of generalized functions

We construct a complete orthonormal system of generalized functions in a Hilbert space W ^?1. We obtain an estimate of the error of approximation in W ^?1, which is expressed in terms of the integral modulus of continuity of a function from L ₂.     

On the Complexity of Self-Validating Numerical Integration and Approximation of Functions with Singularities

We consider the complexity of numerical integration and piecewise polynomial at approximation of bounded functions from a subclass ofC^k([a, b]\Z), whereZis a finite subset of [a, b]. Using only function values or values of derivatives, we usually cannot guarantee that the costs for obtaining an error less thanεare bounded byO(ε^−1/k) and we may have much higher costs. The situation changes if we also allow “realistic” estimates of ranges of functions or derivatives on intervals as observations. A very simple algorithm now yields an error less thanεwithO(ε^−1/k)-costs and an analogous result is also obtained for uniform approximation with piecewise polynomials. In a practical implementation, estimation of ranges may be done efficiently with interval arithmetic and automatic differentiation. The cost for each such evaluation (also of ranges of derivatives) is bounded by a constant times the cost for a function evaluation. The mentioned techniques reduce the class of integrands, but still allow numerical integration of functions from a wide class withO(ε^−1/k) arithmetical operations and guaranteed precisionε.