Sets of multiplicity and differentiable functions. II

Kaufman, Robert

doi:10.1090/S0002-9947-1974-0380258-7

Sets of multiplicity and differentiable functions. II
HTML articles powered by AMS MathViewer

by Robert Kaufman PDF

Trans. Amer. Math. Soc. 200 (1974), 427-435 Request permission

Abstract:

The stability of certain sets of multiplicity is studied with reference to special classes of differentiable functions. Kronecker sets are produced as examples of instability. The most difficult theorem uses probability theory and an estimation of Kolmogoroff’s $\varepsilon$-entropy in a certain space of functions.

References

N. Bourbaki, Éléments de mathématique. I: Les structures fondamentales de l’analyse. Fascicule VIII. Livre III: Topologie générale. Chapitre 9: Utilisation des nombres réels en topologie générale, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1045, Hermann, Paris, 1958 (French). Deuxième édition revue et augmentée. MR 0173226
G. F. Clements, Entropies of sets of functions of bounded variation, Canadian J. Math. 15 (1963), 422–432. MR 158969, DOI 10.4153/CJM-1963-045-3
O. S. Ivašev-Musatov, $M$-sets and $h$-measures, Mat. Zametki 3 (1968), 441–447 (Russian). MR 227689
Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043
Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
R. Kaufman, A functional method for linear sets, Israel J. Math. 5 (1967), 185–187. MR 236607, DOI 10.1007/BF02771106
R. Kaufman, Sets of multiplicity and differentiable functions, Proc. Amer. Math. Soc. 32 (1972), 472–476. MR 340928, DOI 10.1090/S0002-9939-1972-0340928-7
Robert Kaufman, Kronecker sets and metric properties of $M_{0}$-sets, Proc. Amer. Math. Soc. 36 (1972), 519–524. MR 310540, DOI 10.1090/S0002-9939-1972-0310540-4
R. Kaufman, Topics on Kronecker sets, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 4, 65–74 (English, with French summary). MR 340983
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785