Baire’s category theorem and trigonometric series. (English) Zbl 0961.42001

This is an expository article. The subject is Baire’s theorem and its applications to Fourier series, general trigonometric series, Taylor series, Dirichlet series, and thin sets in harmonic analysis.
Baire’s theorem says that in a complete metric space \(X\), any countable intersection of dense open sets \(\{G_n: n= 1,2,\dots\}\) is dense. Its proof is so easy that it has been qualified by T. Körner as a “profound triviality”. On the other hand, difficult constructions can be replaced by a convenient choice of the space \(X\) and the dense open sets \(G_n\). Instead of exhibiting one object \(x\) in \(X\) with a given property \(P\), one proves that \(P\) holds on a countable intersection of open dense subsets of \(X\). In that case, one says that quasi every element \(x\) in \(X\) has property \(P\).
The applications of Baire’s theorem are surprisingly abundant in various areas. We try to illustrate them by presenting a specimen from each of the first four chapters.
Theorem 1.5. Quasi all Fourier-Lebesgue series diverge everywhere.
Corollary 2.8. Quasi all trigonometric series with coefficients tending to \(0\) are universal in the sense of Menshov, that is, every measurable function of \(\mathbb{T}\) can be approximated by some subsequence of the partial sums of the trigonometric series in question almost everywhere on \(\mathbb{T}\).
Denote by \(H(D)\) an appropriate Fréchet space of all analytic functions in the open unit disc \(D\).
Proposition 3.2. For quasi all \(F\in H(D)\), \(F(\Delta)= \mathbb{C}\), where \(\Delta\) is an open set in \(D\) such that the closure \(\overline\Delta\) contains at least a point on the boundary \(|z|= 1\).
Following V. Nestoridis, a series \(S\):\(= \sum^\infty_{n= 0} c_nz^n\), convergent in \(D\) (that is, a Taylor series which represents an analytic function in \(D\)), is called \(H(D)\)-universal if, given any compact set \(K\) such that \(K\cap D= \emptyset\), and any function \(g(z)\) continuous on \(K\) and analytic inside \(K\), there exists a subsequence of the partial sums of \(S\) which converges to \(g(z)\) uniformly on \(K\). Denote by \(A(D)\) the subclass of the functions in \(H(D)\) which represent continuous functions on \(\overline D\).
Theorem 4.2. In \(H(D)\), quasi all Taylor series are universal. In \(A(D)\), quasi all Taylor series are \(A(D)\)-universal; and the same is true in \(A^\infty(D)\) (the space of functions whose derivatives of all orders belong to \(A(D)\)).
A number of very interesting results are included in Ch. 5: Dirichlet series and in Ch. 6: Thin sets, interpolation and superposition.
This beautiful survey article is a must for everyone who wants to keep pace with up-to-date developments in Harmonic Analysis and related fields.


42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
54E52 Baire category, Baire spaces
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[1] N. Yu. Antonov,Convergence of Fourier series, East J. Approx.2 (1996), 187–196. · Zbl 0919.42004
[2] V. Aversa and A. Olevskii,Convergence in category of trigonometric series, Rend. Circ. Mat. Palermo42 (1993), 309–316. · Zbl 0814.42005
[3] N. K. Bari,Trigonometric Series (Russian), Fizmatgiz, Moscow, 1961.
[4] H. Bohr,Collected Mathematical Works (I A1, I A3, III S1, I A2, I A17, I A19, I A18,I A22), Dansk Matematisk Forening, Copenhagen, 1952. · Zbl 0049.00105
[5] P. du Bois Reymond,Ueber die Fourierschen Reihen, Nachr. Kön. Ges. Wiss. Göttingen 21 (1873), 571–582. · JFM 05.0145.01
[6] E. Borel,Sur les séries de Taylor, C.R. Acad. Sci. Paris123 (1896), 1051–1052. · JFM 27.0198.01
[7] L. Carleson,On convergence and growth of partial sums of Fourier series, Acta Math. 116(1966), 133–157. · Zbl 0144.06402
[8] Yung-min Chen,An almost everywhere divergent Fourier series of the class L(log + log + L)1-, J. London Math. Soc.44 (1969), 643–654. · Zbl 0169.39703
[9] C. Chui and M. N. Parnes,Approximation by overconvergence of power series, J. Math. Anal. Appl.36 (1971), 693–696. · Zbl 0224.30006
[10] G. Costakis,Some remarks on universal functions and Taylor series, Math. Proc. Cambridge Philos. Soc, to appear. · Zbl 0956.30003
[11] G. Costakis and A. Melas,On the range of universal functions, submitted. · Zbl 1023.30003
[12] W. Gehlen, W. Luh and J. Müller,On the existence of 0-universal functions, submitted. · Zbl 0805.30002
[13] K-G. Grosse-Erdmann,Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen, Heft 176, ISSN 0373-8221 (1987).
[14] K-G. Grosse-Erdmann,Universal families and hypereyclic operators, Bull. Amer. Math. Soc.36 (1999), 345–381. · Zbl 0933.47003
[15] G.H. Hardy,On the summability of Fourier series, Proc London Math. Soc.12 (1913), 365–372. · JFM 44.0302.01
[16] T. Hedberg,The Kolmogorov superposition theorem, Appendix II in Topics in Approximation Theory, Lecture Notes in Math.187, Springer-Verlag, Berlin, 1971. · Zbl 0219.31012
[17] H. Helson,Fourier transforms on perfect sets, Studia Math.14 (1954), 209–213. · Zbl 0067.33804
[18] R. A. Hunt,On the convergence of Fourier series, inOrthogonal Expansions and their Continuous Analogues (D. T. Haimo, ed.), Southern Illinois Univ. Press, 1968, pp. 235–255.
[19] J.-P. Kahane,Some Random Series of Functions, Heath, Mass., 1968; 2nd edn. Cambridge Univ. Press, 1985, 1993. · Zbl 0192.53801
[20] J.-P Kahane,Séries de Fourier absolument convergentes, Ergebnisse der Mathematik, 50, Springer-Verlag, Berlin, 1970. · Zbl 0195.07602
[21] J.-P. Kahane,Sur les séries de Dirichlet {\(\Sigma\)} {\(\pm\)}n -8 , C.R. Acad. Sci. Paris276 (1973), 739–742.
[22] J.-P Kahane,Sur le théorème de superposition de Kolmogorov, J. Approx. Theory13 (1975), 229–234. · Zbl 0294.26017
[23] J.-P. Kahane,Sur le treizième prohème de Hilbert, le théorème de superposition de Kolmogorov et les sommes algèbriques d’arcs croissants, in Harmonic Analysis Iraklion 1978 Proceedings, Lecture Notes in Math.781, Springer-Verlag, Berlin, 1980, pp. 76–101.
[24] J.-P. Kahane and Y. Katznelson,Sur les ensembles de divergence des sèries trigonomètriques, Studia Math.26 (1966), 305–306. · Zbl 0143.28901
[25] J.-P. Kahane and H. Queffelec,Ordre, convergence et sommabilité des produits de séries de Dirichlet, Ann. Inst. Fourier (Grenoble)47 (1997), 485–529.
[26] J.-P. Kahane and R. Salem,Ensembles parfaits et séries trigonométriques, Paris, Hermann, 1963 (new edition 1985).
[27] Y. Katznelson,An Introduction to Harmonic Analysis, Wiley, New York, 1968; Dover, New York, 1976. · Zbl 0169.17902
[28] R. Kaufman,A functional method for linear sets, Israel J. Math.5 (1967), 185–187. · Zbl 0156.37403
[29] R. Kaufman,Thin sets, differentiable functions and the category method, J. Fourier Anal. Appl. Special Issue Orsay 1993 (1995), 311–316. · Zbl 0927.43001
[30] S. Kierst and E. Szpilrajn,Sur certaines singularités des fonctions analytiques uniformes, Fund. Math.21 (1933), 267–294. · Zbl 0008.07401
[31] A. Kolmogorov (Kolmogoroff),Une série de Fourier–Lebesgue divergente partout, C.R. Acad. Sci. Paris183 (1926), 1327–1328. · JFM 52.0269.02
[32] A. N. Kolmogorov,On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition (in Russian), Dokl. Akad. Nauk SSSR 114(1957), 679–681 ; Engl. transl.: Amer. Math. Soc. Transl.28 (1963), 55–59.
[33] S. V. Konyagin,On divergence of trigonometric Fourier series everywhere, C.R. Acad. Sci. Paris329 (1999), 693–697. · Zbl 0942.42002
[34] P. Koosis,LeÇons sur le théorème de Beurling et Malliavin, Publications du Centre de Recherches Mathématiques, Montréal, 1996. · Zbl 0869.30023
[35] T. Kömer,A pseudofunction on a Helson set I, II, Astérisque5 (1973), 3–224, 231–239. · Zbl 0281.43004
[36] T. Kömer,Kahane’s Helson curve, J. Fourier Anal. Appl. Special Issue Orsay 1993 (1995), 325–346. · Zbl 0886.43008
[37] C. Kuratowski,Topologie, Vol. 1, Monografie matematiczne20, 4th edn., Warszawa, 1958.
[38] W. Luh,Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen88 (1970). · Zbl 0231.30005
[39] W. Luh,Holomorphic monsters, J. Approx. Theory53 (1988), 128–144. · Zbl 0669.30020
[40] S. Mazurkiewicz,Sur l’approximation des fonctions continues d’une variable réelle par les sommes partielles d’une série de puissances, C.R. Soc. Sci. Lett. Varsovie A. III30 (1937), 25–30. · JFM 63.0238.02
[41] A. Melas,On the growth of universal functions, submitted. · Zbl 0973.30002
[42] A. Melas and V. Nestoridis,Universality of Taylor series as a generic property of holomorphic functions, submitted. · Zbl 0985.30023
[43] A. Melas and V. Nestoridis,On various types of universal Taylor series, submitted. · Zbl 1023.30005
[44] A. Melas, V. Nestoridis and I. Papadoperakis,Growth of coefficients of universal Taylor series and comparison of two classes of functions, J. Analyse Math.73 (1997), 187–202. · Zbl 0894.30003
[45] D. E. Mensov (Menchoff),Sur les séries trigonométriques universelles, Dokl. Acad. Sci. SSSR (N.S.)49(1945), 79–82.
[46] D. E. Mensov (Men’shov),On the partial sums of trigonometric series (Russian), Mat. Sb. (N.S.)20(1947), 197–238.
[47] V. Nestoridis,Universal Taylor series, Ann. Inst. Fourier (Grenoble)46 (1996), 1293–1306.
[48] V. Nestoridis,An extension of the notion of universal Taylor series, in Computational Methods and Function Theory ’97 (CMFT’97), Nicoria, Cyprus, 1997, pp. 421–430.
[49] J. O. Oxtoby,Measure and Category, 2nd edn., Springer-Verlag, Berlin, 1980. · Zbl 0435.28011
[50] J. Pál,Zwei Meine Bemerkungen, TÔhoku Math. J.6 (1914-15), 42–43.
[51] R. E. A. C. Paley and A. Zygmund,On some series of functions (1), (2), (3), Proc. Cambridge Philos. Soc.26 (1930), 337–357 ;26 (1930), 458–174;28 (1932), 190–205. · JFM 56.0254.01
[52] H. Queffelec,Propriétés presques sûres et quasi-sûres des séries de Dirichlet et des produits d’Euler, Canad. J. Math.32 (1980), 531–558. · Zbl 0475.30006
[53] A. I. Seleznev,On universal power series (Russian), Math. Sb. (N.S.)28 (1951), 453–460.
[54] P Sjölin,An inequality of Paley and convergence a.e. of Walsh–Fourier series, Ark. Mat.7 (1969), 551–570 · Zbl 0169.08203
[55] H. Steinhaus,über die Wahrscheinlichkeit dafür, dass der konvergenzkreis einer Potenzreihe ihre natürliche Grenze ist, Math. Z. 21 (1929), 408–416. · JFM 55.0187.01
[56] A. G. Vituskin (Vitushkin),On Representations of functions by means of superpostions and related topics, Enseign. Math.23 (1977), 255–310
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