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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.

MSC:

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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