The Uncertainty Principle (up) as understood in this lecture is the following informal assertion: a non-zero “object” (a function, distribution, hyperfunction) and its Fourier image cannot be too small simultaneously. “The smallness” is understood in a very broad sense meaning fast decay (at infinity or at a point, bilateral or unilateral), perforated (or bounded, or semibounded) support etc. The UP becomes a theorem for many “smallnesses” and has a multitude of quite concrete quantitative forms. It plays a fundamental role as one of the major themes of classical Fourier analysis (and neighboring parts of analysis), but also in applications to physics and engineering. The lecture is a review of facts and techniques related to the UP; connections with local and non-local shift invariant operators are discussed at the end of the lecture (including some topical problems of potential theory). The lecture is intended for the general audience acquainted with basic facts of Fourier analysis on the line and circle, and rudiments of complex analysis.

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The uncertainty principle in harmonic analysis

Elements of the Theory of Functions and Functional Analysis, Volume 1. Me and Normed Spaces.

On the universal function for the class Lp[0,1], p ∈ (0,1)

θ-Summation and Hardy Spaces

A general summability method of Fourier series and Fourier transforms is given with the help of an integrable function % having integrable Fourier transform. Under some weak conditions on % we show that the maximal operator of the %-means of a distribution is bounded from Hp(T )t oLp(T )( p0<p<) and is of weak type (1,1), where Hp(T) is the classical Hardy space and p0<1 is depending only on %. As a consequence we obtain that the %-means of a function f # L1(T) converge a.e. to f. For the endpoint p0 we get that the maximal operator is of weak type (Hp0(T), Lp0(T)). Moreover, we prove that the %-means are uniformly bounded on the spaces Hp(T) whenever p0<p< and are uniformly of weak type (Hp0(T), Hp0(T)). Thus, in the case f # Hp(T), the %-means converge to f in Hp(T) norm ( p0<p<). The same results are proved for the conjugate %-means and for Fourier transforms, too. Some special cases of the %-summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, Riemann, de La Vallee-Poussin, Rogosinski and Riesz summations. 2000 Academic Press

Summation and Hardy Spaces 1

By a general Franklin system corresponding to a dense sequence T = (tn, n ≥ 0) of points in [0, 1] we mean a sequence of orthonormal piecewise linear functions with knots T , that is, the nth function of the system has knots t0, . . . , tn. The main result of this paper is that each general Franklin system is an unconditional basis in L[0, 1], 1 < p <∞.

Unconditionality of general Franklin systems in $L^{p}[0,1]$, 1 < p < ∞

It is proved that (1)is the Fourier series of an integrable function if and only if1) almost everywhere, and2) \lambda\,\}=0$ SRC=http://ej.iop.org/images/0025-5734/68/2/A02/tex_sm_2107_img4.gif/>,where 0}\vert S(x,\,h)\vert.$ SRC=http://ej.iop.org/images/0025-5734/68/2/A02/tex_sm_2107_img5.gif/>Bibliography: 6 titles.

On the uniqueness of trigonometric series

We prove that if and then the Walsh series has the following property. For any measurable function which is finite almost everywhere, there are numbers such that the series converges to almost everywhere. This assertion complements and strengthens previously known results about universal Walsh series and Walsh null-series.

https://iopscience.iop.org/article/10.1070/IM1999v063n01ABEH000227/pdf

On Walsh series with monotone coefficients

By a general Franklin system corresponding to a dense sequence of knots T = (tn;n 0) in (0; 1) we mean a sequence of orthonormal piecewise linear functions with knotsT , that is, the nth function of the system has knots t0;:::;tn. The main result of this paper is a characterization of sequences T for which the corresponding general Franklin system is a basis or an unconditional basis in H 1 (0; 1).

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General Franklin systems as bases in H¹[0,1]

Trigonometric series summable by the Riemann method almost everywhere are considered. In particular, it is proved that if a multiple trigonometric series sums almost everywhere by the Riemann method to an integrable function f(x), and the Riemann majorant of this series satisfies a certain necessary condition, then the series is the Fourier series of the function f(x).Bibliography: 11 titles.

https://iopscience.iop.org/article/10.1070/SM1995v080n02ABEH003528/pdf

G. G. Gevorkyan

Papers

Unconditionality of general Franklin systems in $L^{p}[0,1]$, 1 < p < ∞

On the uniqueness of trigonometric series

On Walsh series with monotone coefficients

General Franklin systems as bases in H¹[0,1]

On uniqueness of multiple trigonometric series