By Dengfeng L., Silong P.

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**Extra info for A Caracterization of N-dimensional Daubechies Type Tensor Product Wavelet**

**Example text**

For a recent account of these methods, we refer to [13]. Another common principle requires the image representation to be sparse. In our particular situation, this means that the triangulation should be as small as possible. Note that the requirement of sparsity is not necessarily reﬂected by the number of vertices (or edges or triangles) in the triangulation, but it can also be characterized by some suitable entropy measure related to a compression scheme. The construction of sparse triangulations for edge-adapted image approximation, according to the above construction principles, leads to an abstract approximation problem, whose general framework is brieﬂy introduced in Section 2.

1)d . Clearly, Algorithms 2 and 3 produce a complete subtree T of T dc in the sense that for any node in T its parent and all siblings are also in T . The corresponding partition Δ consists of all leaves of T . If we set e(ω) = f − fω pLp (ω) , then E(T ) := ω∈Δ e(ω) = f − sΔ (f ) pp . It is easy to see that |Δ| = 1 + (2d − 1)n(T ), where n(T ) denotes the number of subdivisions used to create T . The quantity n(T ) measures the complexity of a tree, and En := inf n(T )≤n E(T ) gives the optimal error achievable by a tree of a given complexity.

Section 2 is devoted to a simple linear approximation algorithm based on a uniform subdivision of the domain and local approximation by constants. In addition, we show that the approximation order N −1/d cannot be improved on isotropic partitions and give a review of the results on the approximation by constants (Sobolev-Poincar´e inequalities) on general domains. Section 3 is devoted to the methods of nonlinear approximation restricted to our topic of piecewise constants. We discuss adaptive partition based methods such as Birman-Solomyak’s algorithm and tree approximation, as well as dictionary based methods such as Haar wavelet Piecewise Constant Approximation 29 thresholding and best n-term approximation.

### A Caracterization of N-dimensional Daubechies Type Tensor Product Wavelet by Dengfeng L., Silong P.

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