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Approximation of Functions

Module by: Veronique Delouille. E-mail the author

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We first give a definition of the order of a multiresolution analysis.

Definition:

(Order of a MRA in the classical setting) A multiresolution analysis is said to be of order N˜N˜ if the primal scaling function ϕϕ reproduces polynomials up to degree N˜-1N˜-1, i.e., For 0 ≤ p < N ˜ , ∃ c k ∈ R such that x p = ∑ k c k ϕ ( x - k ) . For 0 ≤ p < N ˜ , ∃ c k ∈ R such that x p = ∑ k c k ϕ ( x - k ) .

The associated dual wavelet ψ˜ψ˜ has then N˜N˜ vanishing moments. In the classical setting, it is proved that the order of a MRA and the regularity of the scaling function are linked: the larger N˜N˜, the higher the regularity of ϕϕ. Symmetrically to Definition 1, the order of the dual MRA is NN if ϕ˜ϕ˜ can reproduce polynomials up to degree N-1N-1. Figure 2 from Multiresolution analysis and wavelets shows an example of a biorthogonal basis where N˜=3N˜=3 and N=1N=1. It illustrates the link between a high number of vanishing moments of the dual wavelet ψ˜ψ˜ and the regularity of the primal scaling function ϕϕ.

The main objective when decomposing a function in a wavelet series is to create a sparse representation of the function, that is, to obtain a decomposition where only a few number of detail coefficients are `large', while the majority of the coefficients are close to zero. By `large', we mean that the absolute value of the detail coefficient is large.

Near a singularity, large detail coefficients at different levels will be needed to recover the discontinuity. However, between points of singularity, we can hope to have small detail coefficients, in particular if the analyzing wavelets ψ˜jkψ˜jk have a large number N˜N˜ of vanishing moments. Indeed, suppose the function ff to be decomposed is analytic on the interval II without discontinuity. Since xp,ψ˜jk=0xp,ψ˜jk=0 for p=0,...,N˜-1p=0,...,N˜-1, we are sure that the first N˜N˜ terms of a Taylor expansion of ff will not give a contribution to the wavelet coefficient f,ψ˜jkf,ψ˜jk provided that the support of ψ˜jkψ˜jk does not contain any singularities of the function ff.

This sparse representation explains why classical wavelets provide smoothness characterization of function spaces like the Hölder and Sobolev spaces [2], but also of more general Besov spaces, which may contain functions of inhomogeneous regularity [7], [5], [4], [3], [6].

We illustrate this characterization property with the case of β-β-Hölder functions.

Definition 2

The class Λβ(L)Λβ(L) of Hölder continuous functions is defined as follows:

  1. if β≤1,Λβ(L)=f:f(x)-f(y)≤L|x-y|ββ≤1,Λβ(L)=f:f(x)-f(y)≤L|x-y|β
  2. if β>1,Λβ(L)=f:f⌊β⌋(x)-f⌊β⌋(y)≤L|x-y|β';|f⌊β⌋(x)|≤M,β>1,Λβ(L)=f:f⌊β⌋(x)-f⌊β⌋(y)≤L|x-y|β';|f⌊β⌋(x)|≤M, where ⌊β⌋⌊β⌋ is the largest integer less than ββ and β'=β-⌊β⌋.β'=β-⌊β⌋.

The global Hölder regularity of a function can be characterized as follows [1], [2].

Theorem:

Let f∈Λβ(L)f∈Λβ(L), and suppose that the (orthogonal) wavelet function ψψ has rr continuous derivatives and rr vanishing moments with r>βr>β. Then
f , ψ j k ≤ C 2 - j ( β + 1 / 2 ) . f , ψ j k ≤ C 2 - j ( β + 1 / 2 ) . (1)

A similar characterization exists for continuous and Sobolev functions [2], [7].

In the orthogonal setting, the wavelet ψψ must be regular and have a high number of vanishing moments. On the contrary, in the biorthogonal expansion equation 5 from Multiresolution analysis and wavelets, it is mostly of interest to have a dual wavelet ψ˜ψ˜ with a high number of vanishing moments, and hence a regular primal scaling and wavelet functions. On the primal side, it is sufficient to have only one vanishing moment for wavelet denoising, and consequently ψ˜ψ˜ may not be very regular. In this case, the wavelet coefficient f,ψ˜jkf,ψ˜jk with the less regular wavelet ψ˜jkψ˜jk can be used to characterize f∈Λβ(L)f∈Λβ(L) with 0<β<N˜0<β<N˜, even if β>N=1β>N=1: with a biorthogonal basis, regular functions can be characterized by their inner products with much less regular functions.

References

  1. Cai, T. and Brown, L.D. (1998). Wavelet Shrinkage for nonequispaced samples. Annals of Statistics, 26(5), 1783-1799.
  2. Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: SIAM.
  3. DeVore, R.A. and Jawerth, B.B. and Popov, V. (1992). Compression of wavelet decomposition. Amer. J. Math, 114, 737-785.
  4. Donoho, D.L. (1992). Interpolating wavelets. Preprint. Department of Statistics, Stanford University.
  5. Donoho, D.L. (1995). De-noising via soft-thresholding. IEEE Transactions on Information Theory, 41, 613-627.
  6. DeVore, R.A. and Popov, V. (1988). Interpolation of Besov spaces. Trans. Amer. Math. Soc., 305, 397-414.
  7. Härdle, W. and Kerkyacharian, G. and Picard, D. and Tsybakov, A. (1998). Lecture Notes in Statistics 129: Wavelets, Approximation, and Statistical Applications. Springer-Verlag.

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