Any finite union of intervals supports a Riesz basis of exponentials

Analysis 2 Minutes

You need to know: Notations {\mathbb Z}, {\mathbb R}, and {\mathbb C} for sets of integer, real, and complex numbers, respectively. Notation i for complex number \sqrt{-1}, absolute value |z| and conjugate \bar{z} of z\in{\mathbb C}, exponent of a complex number, integral, sum of infinite series.

Background: Let S\subseteq {\mathbb R} be a set (for this Theorem, we only need the case when S in a finite union of intervals). Let L^2(S) be the set of functions f:S \to {\mathbb C} for which the integral \int_S |f(x)|^2dx exists and finite. For f,g \in L^2(S), denote (f,g)=\int_S f(x)\bar{g}(x)dx and \|f\|^2=(f,f). A sequence \{f_n\}_{n=1}^\infty of functions in L^2(S) is called complete if the only f \in L^2(S) satisfying (f,f_n)=0 for all n is f=0. A complete sequence \{f_n\}_{n=1}^\infty is called a Riesz basis if there exist positive constants c and C such that c\sum\limits_{n=1}^\infty |a_n|^2 \leq \|\sum\limits_{n=1}^\infty a_n f_n\| \leq C\sum\limits_{n=1}^\infty |a_n|^2 for every sequence \{a_n\}_{n=1}^\infty of complex numbers such that \sum\limits_{n=1}^\infty |a_n|^2 < \infty.

The Theorem: On 23th October 2012, Gady Kozma and Shahaf Nitzan submitted to arxiv a paper in which they proved that, whenever S\subseteq {\mathbb R} is a finite union of intervals, there exists a set \Lambda\subset{\mathbb R} such that the functions \left\{e^{i \lambda t}\right\}_{\lambda \in \Lambda} form a Riesz basis in L_2(S). Moreover, if S \subseteq [0, 2\pi] then \Lambda may be chosen to satisfy \Lambda \subseteq {\mathbb Z}.

Short context: A Riesz basis of exponential functions as in the Theorem gives each function f \in L^2(S) a unique representation f (t) = \sum c_{\lambda} e^{i\lambda t}, which is useful for many applications. However, there are relatively few examples of sets S for which such basis is known to exists. For example, it was known that it exists if S is an interval. The Theorem extends this to the important case when S is a finite union of intervals.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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