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
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(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:
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∫
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g
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.
(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 :
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(3)-
C.K. Chui. (1992). An Introduction to Wavelets. San Diego, CA: American Press.
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I. Daubechies. (1992). Ten Lectures on wavelets. Philadelphia: Society for Industrial and Applied Mathematics.
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W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybekov. (1998). Wavelets, Approximation, and Statistical Applications. Lectures notes in Statistics.
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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.
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Y. Meyer. (1990). Ondelettes et Opérateurs, I: Ondelettes, II:. Operateurs de Calderón-Zygmund.
