Download PDF by S. Kotz: Encyclopedia of Statistical Scis. [Vol. 15]

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Donoho, D. L. (1996). De-noising by softthresholding. IEEE Trans. Inf. Theory, 41, 613–627. 33. Donoho, D. L. (1997). CART and Best-OrthoBasis: A Connection. Ann. , 25, 1870–1911. 34. Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet–vaguelette decomposition. Appl. Comput. , 2, 101–126. 35. Dubuc, S. (1986). Interpolation through an iterative scheme. J. Math. Anal. , 114, 185–204. 36. Efromovich, S. and Pinkser, M. (1981). Estimation of square-integrable density on the basis of a sequence of observations.

When generating orthonormal wavelet bases, we also require that the space V0 of the MRA contain a function φ such that the integer translations {φ0,n } form an orthonormal basis for V0 . For simplicity, we will focus almost exclusively on orthonormal bases here. Since V0 ⊂ V1 , we can define its orthogonal complement W0 in V1 , so V1 = V0 ⊕ W0 . We can do likewise for every j, where Vj+1 = Vj ⊕ Wj . , f ∈ Wj if and only if f (2−j ·) ∈ W0 . By themselves, these spaces provide a homogeneous representation of L2 (R), since L2 (R) = clos {∪j∈Z Wj }.

Lp , Sobolev, H¨older, and Zygmund spaces) and many other spaces containing functions with interesting and quite distinctive structure. The work of Donoho and Johnstone, described in the next section, contains a discussion of the importance of these spaces. The unconditional basis property explains many of the most important properties of wavelet bases, in particular the tendency of wavelet expansions to be sparse and the effectiveness of procedures that act on individual wavelet coefficients.

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Encyclopedia of Statistical Scis. [Vol. 15] by S. Kotz

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