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Let F be a compact set of a Banach space X. This paper analyzes the "generalized empirical interpolation method," which, given a function f ∊ F, builds an interpolant Jn[f] in an n dimensional subspace Xn ⊂ X with the knowledge of n outputs (σi(f))ni=1, where σi ∊ X' and X' is the dual space of X. The space Xn is built with a greedy algorithm that is adapted to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ ⊂ X'. In this paper, we study the interpolation error maxf∊F ‖f - Jn[f]‖X by comparing it with the best possible performance on an n dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper we study a variational approach for filling in regions of missing data in two-dimensionaland three-dimensionaldigitalimages. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity, or the zooming of images. The approach presented here, initially introduced in [IEEE Trans. Image Pr cess.,10 (2001), pp. 1200-1211] is based on a joint interpolation of the image gray levels and gradient/isophotes directions, smoothly extending the isophote lines into the holes of missing data. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. We study the existence of minimizers of our functional and its approximation by minima of smoother functionals. Then we present the numerical algorithm used to minimize it and display some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we consider a method of adapting Dubuc -Deslauriers subdivision, which is defined for bi-infinite sequences, to accommodate sequences of finite length. After deriving certain useful properties of the Dubuc -Deslauriers refinable function on R, we define a multiscale finite sequence of functions on a bounded interval, which are then proved to be refinable. Using this fact, the resulting adapted interpolatory subdivision scheme for finite sequences is then shown to be convergent. Corresponding interpolation wavelets on an interval are defined, and explicit formulations of the resulting decomposition and reconstruction algorithms are calculated. Finally, we give two numerical examples on signature smoothing and two-dimensional feature extraction of the subdivision and wavelet algorithms. [ABSTRACT FROM AUTHOR]