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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.

##### MSC:
 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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