Wavelets on fractals and Besov spaces
The author develops the theory of wavelets on self-similar fractals \(K\) which preserves a certain standard inequality (Markov's inequality). This property is equivalent to the statement that for all balls \(B\) centered at a point in \(K\), and all polynomials \(P\) of degree at most \(m\), one has \[ \sup_{x \in B} | P(x)| \leq C_m \sup_{x \in B \cap K} | P(x)| . \] For such a fractal \(K\) and a positive integer \(m\), an orthonormal wavelet basis is constructed; the basis elements are compositions of piecewise polynomial functions of degree at most \(m\) with the similitudes of \(K\). This wavelet basis is then shown to characterise the Besov spaces \(B^{p,q}_\alpha(K)\) in the expected manner. In particular, if \(\dim(K) = s\) and \(f\) is a function in the Lipschitz space \(\Lambda_\alpha(K)\) for some \(\alpha > 0\), then the wavelet coefficients at scale \(i\) decay like \(\text{diam} (K_i)^{\alpha+s/2}\) providing that \(m \geq {\alpha}\), where \(i\) is a sequence of similitudes and \(K_i\) is the image of \(K\) under this sequence. The converse is also true if the \(K_i\) are disjoint; this statement does not require \(m \geq {\alpha}\).
scientific article; zbMATH DE number 1220781
scientific article