On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.

Dual Algebras and A-Measures

Weak-star closures of Gleason parts in the spectrum of a function algebra are studied. These closures relate to the bidual algebra and turn out both closed and open subsets of a compact hyperstonean space. Moreover, weak-star closures of the corresponding bands of measures are reducing. Among the applications we have a complete solution of an abstract version of the problem, whether the set of nonnegative A-measures (called also Henkin measures) is closed with respect to the absolute continuity. When applied to the classical case of analytic functions on a domain of holomorphyΩ⊂Cn, our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity ofΩ. We also investigate the relation between the algebra of bounded holomorphic functions onΩand its abstract counterpart—thew* closure of a function algebraAin the dual of the band of measures generated by one of Gleason parts of the spectrum ofA.

Rapid convergence of some seismic processing algorithms

The rate of convergence of many numerical algorithms can be greatly improved by repeated application of the method of summation by parts. Cases of interest in seismology arise when we need to resample the spectrum of a function at unevenly spaced frequency values. Related examples include the time‐domain evaluation of the Hilbert transform and the extrapolation to the real axis of spectra evaluated in the complex domain. A formula of this type was first presented by Lanczos (1956). The validity of the algorithms depends upon the fact that summation by parts of the cardinal‐series representation is justified as long as a function is somewhat oversampled.