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, Laboratoire d'Analyse Numérique
, Nous présentons une méthode de quadrature, pour le calcul du produit scalaire / = / f(t)é(t)dt, lorsque /(r) présente une singularité de type homogène, la fonction <p(t) étant définie par une équation d'échelle. Singular quadratures and scaling junctions
, A quadrature method is presented for the computation of the inner product. I = J f{t)<p{t)dt, where f(t) has a singularity of homogeneous type
, 3 . ? k) is the generator of the underlying multiresolution approximation Vj. The difficulties in computing these quantities arise from the fact the / and K might be singular and <j> is only known implicitely as a solution of a refinement equation 4>(x) = 2 Ylk=o hk<f>(2% -£)? As a prototype example, we consider the computation oí I = J f(t)<f>{t)dt. In the case where the function / is C', quadrature rules have been introduced in [1], [9] using the fact that the moments Mk = J x k d>(x)dx can easily be computed from the refinement equation. A first approximation of I is then obtained by replacing / by a polynomial interpolation of degree 1-1 on the support of </>. In order to improve the precision, one can use a subdivision method: iterating the refinement equation, Abridged English Version Wavelet bases have proved to be a powerful tool for the discretization of singular integral equations, since they usually lead to sparse matrices. However one obtains <p(x) ? J3 fc Sj i k 2 i <j>(2^x ? fc), and the initial quadrature rule can be replaced by a combination of similar rules at scale 2 _J . An analysis of the coefficients SJ_ k and of the local error, shows that the numerical precision is of order 2~-> l, pp.2-4
, Ezzine Note remise et acceptée le 11 mai 1996
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