Natalia Kholshchevnikova

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Speaker

Natalia Nikolaevna Kholshchevnikova is a plenary speaker, professor of mathematics at the Moscow State Technological University "Stankin".

Natalia graduated from Lomonosov Moscow State University, defended her candidate and then doctoral dissertations at the Mathematical Institute of RAS. Her research interests lie in the theory of functions, namely in the theory of uniqueness and representation of functions by series and the study of set-theoretic properties of thin sets. She was a participant of several RFBR grants, INTAS grant, a short-term Soros grant, and received a State scientific scholarship. She has participated in many international conferences.

Lecture

The Union Problem and the Category Problem of Sets of Uniqueness in the Theory of Orthogonal Series

The main impulse to the development of the theory of uniqueness for trigonometric series was the G.Cantor theorem (1870): If trigonometric series converges everywhere to zero then all coefficients of this series are equal zero; and the example of D. E. Menshov (1916), who constructed trigonometric series that converges to zero almost everywhere but not everywhere.

Definitions. A set [math]E\subset \mathbb{T}[/math] is called a set of uniqueness or [math]U[/math]-set if every trigonometric series which converges to zero outside [math]E[/math] has all coefficients equal zero. Otherwise [math]E[/math] is called a set of multiplicity or an [math]M[/math]-set.

Cantor proved that empty and also finite and some countable sets are [math]U[/math]-sets. A.Rajchman (1922) and N.K.Bary (1923) constructed perfect [math]U[/math]-sets. Then Union and Category problems aroused.

The Union Problem

Is the union of two (countable many) "good" (Borel or analytic) sets of uniqueness also a [math]U[/math]-set?

The famous Theorem was proved by N.K.Bary (1923):

The union of countably many closed sets of uniqueness is a set of uniqueness.

In 1981 N.N.Kholshchevnikova proved that:

  1. The union of two disjoint [math]G_\delta[/math] sets of uniqueness is a set of uniqueness.
  2. The union of two sets of uniqueness A and B, where B is both a [math]G_\delta[/math] and an [math]F_\sigma[/math], is a set of uniqueness.

C.Carlet and G.Debs (1985) generalised these results: The union of a sequence of U-sets [math]E_n[/math] which are relatively closed in their union [math]\cup_{n=1}^{\infty}E_n[/math] is a [math]U[/math]-set.

The Union Problem is open for two [math]G_\delta[/math] [math]U[/math]-sets and for [math]G_\delta[/math] [math]U[/math]-set and a countable set. For Walsh system, analogous results were obtained by A.Shneider(1947), W.R.Wade(1971), Kholshchevnikova(1992).

The Category Problem

Is every Borel U-set of the first category?

This problem for trigonometric series was solved positively by G.Debs and J.Saint Raymond (1986), and for Walsh series by Kholshchevnikova (1993).

Trigonometric and Walsh systems are systems of characters on a compact abelian group. The theory of uniqueness for systems of characters on a zero-dimensional compact abelian group with the second axiom of countability is actively developed now. V.A.Skvortsov and Kholshchevnikova (2017) solved affirmatively the category problem for such systems and obtained a generalization of the theorem on existence of a perfect [math]M_0[/math]-set whose Hausdorff [math]h[/math]-measure equals zero inside of any closed [math]M_0[/math]-set.

Many interesting results were obtained in the theory of uniqueness for multiple series (trigonometric, Walsh, Haar and other) and for the different type of convergence (rectangles, cubes, spheres, etc). In particular, many results that we can relate to the Union Problem were obtained by A.A.Talalyan, F.G.Arutyunyan, S.F.Lukomsky, J.M.Ash, G.V.Welland, G.Wang, Sh.T.Tetunashvili, J.Bourgain, V.A.Skvortsov, N.A.Bokaev, N.N.Kholshevnikova, M.G.Plotnikov, T.A.Sworovska, L.D.Gogoladze.