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On the Frisch-Parisi conjecture. (English) Zbl 0963.28009
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.

28A80 Fractals
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
28A78 Hausdorff and packing measures
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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