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Quadrature formulas based on the scaling function

Václav Finěk (2005)

Applications of Mathematics

The scaling function corresponding to the Daubechies wavelet with two vanishing moments is used to derive new quadrature formulas. This scaling function has the smallest support among all orthonormal scaling functions with the properties M 2 = M 1 2 and M 0 = 1 . So, in this sense, its choice is optimal. Numerical examples are given.

Quantitative properties of quadratic spline wavelet bases in higher dimensions

Černá, Dana, Finěk, Václav, Šimůnková, Martina (2015)

Programs and Algorithms of Numerical Mathematics

To use wavelets efficiently to solve numerically partial differential equations in higher dimensions, it is necessary to have at one’s disposal suitable wavelet bases. Ideal wavelets should have short supports and vanishing moments, be smooth and known in closed form, and a corresponding wavelet basis should be well-conditioned. In our contribution, we compare condition numbers of different quadratic spline wavelet bases in dimensions d = 1, 2 and 3 on tensor product domains (0,1)^d.

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