[en] We give three characterizations of the elements of generalized Hölder-Zygmund spaces. The first one, based on the Littlewood-Paley decomposition is already known, but the proof given here is much simpler. The second one, based on the wavelet decompositions generalizes a result obtained by Jaffard and Meyer. The third one uses generalized interpolation spaces. These results naturally extend the ones holding for the classical Hölder-Zygmund spaces.
The manuscript has been accepted for publication in Journal of Approximation Theory.
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Bibliography
Almeida Alexandre Wavelet representation in besov spaces of generalized smoothness. Function Spaces, Differential Operators and Nonlinear Analysis 2005, 7-18. P. Drábek, J. Rákosník (Eds.).
Besov Oleg Vladimirovich, Il'in Valentin Petrovich, Nikol'skii Sergey Mikhaylovich Integral Representation of Functions and Imbeddings Theorems, Vol. I, Mitchell H. Taibleson 1978, John Wiley and sons edition, Washington, DC.
Caetano António M., Moura Susana D. Local growth envelopes of spaces of generalized smoothness: the subcriticalcase. Math. Nachr. 2004, 273(1):43-57.
Daubechies Ingrid Ten Lectures on Wavelets 2006, Society for Industrial and Applied Mathematics.
Farkas Walter, Leopold Hans-Gerd Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl. 2006, 185(1):1-62.
Fekete Michael On the distribution of roots of algebraic equations with integral coefficients. Math. Z. 1923, 17:228-249. (Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten).
Stéphane Jaffard, Wavelet techniques in multifractal analysis, in: Proceedings of Symposia in Pure Mathematics, vol. 72, 2004, pp. 91-152.
Jaffard Stéphane, Meyer Yves Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions. Mem. Am. Math. Soc. 1996, vol. 587:110.
Kreit Damien, Nicolay Samuel Some characterizations of generalized Hölder spaces. Math. Nachr. 2012, 285:2157-2172.
Mallat Stéphane A Wavelet Tour of Signal Processing 1999, Academic Press, San Diego.
Meyer Yves Ondelettes et Opérateurs: Ondelettes, vol. 1 1990, Hermann.
Moura Susana D. On some characterizations of Besov spaces of generalized smoothness. Math. Nachr. 2007, 280(9-10):1190-1199.
Pilipović Stevan, Rakić Dušan, Vindas Diaz Jasson New classes of weighted Hölder-Zygmund spaces and the wavelet transform. J. Funct. Spaces Appl. 2012, 2012:18.
Rauhut Holger, Ullrich Tino Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type. J. Funct. Anal. 2011, 260(11):3299-3362.
Stein Elias M. Singular Integrals and Differentiability Properties of Functions, vol. 30 1970, Princeton Univ. Pr.
Stein Elias M. Topics in Harmonic Analysis Related to the Littlewood-Paley Theory 1970, Princeton University Press.
Tartar Luc An Introduction to Sobolev Spaces and Interpolation Spaces, vol. 3 2007, Springer Verlag.
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