Differential equations by Bateman H.

Differential equations by Bateman H.

By Bateman H.

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Before we proceed, let us introduce an abbreviation that will frequently be used in the sequel. g. 11) the constant is independent of j and s). A B is defined in a similar way and A ∼ B means that A B and A B with possibly different constants, of course. 2 Piecewise linear systems It is a well-known fact that one can obtain a better rate of approximation by using more regular functions than piecewise constants. 4 Approximation error for fJ for the supremum and Euclidean norm. 11) is caused by the fact that we consider only piecewise constant approximations.

8 (first row 1, . . , 4, second row 5, . . , 8 from left to right). This was an open problem for quite some time, and was finally solved by Ingrid Daubechies [96, 97]. Nowadays, these functions are widely used in various applications. There exists a whole family of such functions labeled by a parameter N , which roughly means that the functions are smoother (and less local) for increasing values of N . There are no closed formulas for the arising functions N ϕ. The functions are only given in terms of their refinement coefficients.

The road map is as follows: Using wavelets allows us to reformulate the differential equation as a discrete infinite-dimensional problem for the unknown wavelet expansion coefficients of the solution. Moreover, this discrete problem is well conditioned. An adaptive scheme results by first considering an infinitedimensional iteration and then a computable version arises by introducing finite approximations of the involved operators. We introduce the analysis of these schemes and describe numerical experiments.

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