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Preliminaries

Module by: Veronique Delouille

We begin by recalling some basic notions of functional analysis. A measurable function ff belongs to the Lebesgue space Lp(IR),1≤p<∞Lp(IR),1≤p<∞ if

| | f | | p = ( ∫ - ∞ + ∞ | f ( x ) | p d x ) 1 / p < ∞ . | | f | | p = ( ∫ - ∞ + ∞ | f ( x ) | p d x ) 1 / p < ∞ . (1)

A Hilbert space is a space where an inner product is defined. In particular, the space L2(IR)L2(IR) is a Hilbert space, where the inner product of 2 functions ff and gg is defined as:

< f , g > = ∫ - ∞ + ∞ f ( x ) g ( x ) ¯ d x . < f , g > = ∫ - ∞ + ∞ f ( x ) g ( x ) ¯ d x . (2)

In this presentation we work with functions defined on IR,IR, but that take values in C[0.1ex]0.05em1.25ex.C[0.1ex]0.05em1.25ex. Hence g(x)¯g(x)¯ denotes the complex conjugate of g(x).g(x). We say that 2 functions are orthogonal if their inner product is zero. A function is Hölder continuous of order α,(0<α≤1)α,(0<α≤1) at point xx if :

| f ( x ) - f ( x + h ) | = O ( h α ) . | f ( x ) - f ( x + h ) | = O ( h α ) . (3)

References

  1. C.K. Chui. (1992). An Introduction to Wavelets. San Diego, CA: American Press.
  2. I. Daubechies. (1992). Ten Lectures on wavelets. Philadelphia: Society for Industrial and Applied Mathematics.
  3. W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybekov. (1998). Wavelets, Approximation, and Statistical Applications. Lectures notes in Statistics.
  4. S. Mallat. (1989). A theory for multiresolution signal decomposition: the wavelet representation. I.E.E.E Trans. on pattern analysis and machine intelligence, 11(7), 674-693.
  5. Y. Meyer. (1990). Ondelettes et Opérateurs, I: Ondelettes, II:. Operateurs de Calderón-Zygmund.

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