## Computation with wavelets in higher dimensions.(English)Zbl 0907.42023

Summary: In dimension $$d$$, a lattice grid of size $$N$$ has $$N^d$$ points. The representation of a function by, for instance, splines or the so-called non-standard wavelets with error $$\varepsilon$$ would require $$O(\varepsilon^{-ad})$$ lattice point values (resp. wavelet coefficients), for some positive $$a$$ depending on the spline order (resp. the properties of the wavelet). Unless $$d$$ is very small, we easily will get a data set that is larger than a computer in practice can handle, even for very moderate choices of $$N$$ or $$\varepsilon$$.
I discuss how to organize the wavelets so that functions can be represented with $O((\log(1/\varepsilon))^{a(d- 1)}\varepsilon^{- a})$ coefficients. Using wavelet packets, the number of coefficients may be further reduced.

### MSC:

 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 65D07 Numerical computation using splines
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