We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p <2, so our method can signi cantly outperform every linear method (kernel, smoothing spline, sieve,:::) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which mayvary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems.

An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an unconditional basis works essentially better for recovery and estimation than other methods, period. (Performance is measured in an asymptotic minimax sense.) As an application, we formalize and prove Mallat's Heuristic, which says that wavelet bases are optimal for representing functions containing singularities, when there may be an arbitrary number of singularities, arbitrarily distributed.

We investigate the order of convergence of periodic interpolation on sparse grids (blending interpolation) in the framework of Nikol'skij-Besov spaces of dominating mixed smoothness. The main ingredients are a unified approach to error estimates in univariate Nikol'skij-Besov spaces and the tensor product characterization of the bivariate Nikol'skij-Besov spaces of dominating mixed smoothness.

Using periodic Strang--Fix conditions, we can give an approach to error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces. 1 Introduction We investigate the L 2 --error of interpolation on equidistant and sparse grids for periodic functions from isotropic L 2 --Besov spaces and L 2 --Besov spaces of functions with dominating mixed smoothness properties. The interpolation of periodic functions by translates of a given function and the corresponding error estimates have been analyzed by several authors (e.g. [3, 8, 14]) in the univariate as well as in the multivariate case. The periodic Strang--Fix conditions were introduced in [2, 14]. There, they were used to find L 2 --error estimates for functions from isotropic L 2 --Sobolev spaces. The approximation of functions on sparse grids and the related field of hyperbolic approximation have a fairly long tradition (e.g. [4, 5, 27]) as well. For bivariate functions, the number of i...

properties for some scales of approximation spaces