On the Frisch-Parisi conjecture
Let \(x_0 \in {\mathbb R}^d\) and let \(\alpha > 0\) be a real number. A function \(f: {\mathbb R}^d \to {\mathbb R}\) is in \(C^{\alpha}(x_0)\) if there exists a constant \(C>0\) and a polynomial \(P_{x_0}\) of degree at most \([\alpha]\) such that in a neighborhood of \(x_0\), \[ |f(x) - P_{x_0}(x)|\leq C |x - x_0|^{\alpha}. \] The Hölder exponent of \(f\) at \(x_0\) is defined by \[ h_f(x_0) = \sup\{\alpha> 0: f \in C^{\alpha}(x_0)\}. \] Let \(S_H = \{x: h_f(x) = H\}\). Then \(d(H) = \dim(S_H)\) is called the spectrum of singularities of \(f\). A function is called multifractal when its spectrum of singularities is defined at least on an interval of non-empty interior. In this paper, the author proves several results on the genericity (in the sense of Baire's categories) of multifractal functions. Among other results, he proves that if \(p > 0\), \(q > 0\) and \(s > d/p\), then quasi-all functions of the Besov space \({\mathbf B}_p^{s, q}({\mathbb R}^d)\) are multifractal functions. The domain of definition of the spectrum of singularities is the interval \([s-d/p,\;s]\) on which the spectrum is given by \[ d(H) = p(H-s) + d. \] If \(p > 1\) the same is true for the Sobolev space \(L^{p,s}({\mathbb R}^d)\). He also shows that the Frisch-Parisi conjecture holds for quasi-all functions. The main tool for proving these results is orthonormal wavelet decomposition.
scientific article; zbMATH DE number 1522078