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The present book is a collection of variations on a theme which can be summed up as follows: It is impossible for a non-zero function and its Fourier transform to be simultaneously very small. In other words, the approximate equalities x :::::: y and x :::::: fj cannot hold, at the same time and with a high degree of accuracy, unless the functions x and yare identical. Any information gained about x (in the form of a good approximation y) has to be paid for by a corresponding loss of control on x, and vice versa. Such is, roughly speaking, the import of the Uncertainty Principle (or UP for short) referred to in the title ofthis book. That principle has an unmistakable kinship with its namesake in physics - Heisenberg's famous Uncertainty Principle - and may indeed be regarded as providing one of mathematical interpretations for the latter. But we mention these links with Quantum Mechanics and other connections with physics and engineering only for their inspirational value, and hasten to reassure the reader that at no point in this book will he be led beyond the world of purely mathematical facts. Actually, the portion of this world charted in our book is sufficiently vast, even though we confine ourselves to trigonometric Fourier series and integrals (so that "The U. P. in Fourier Analysis" might be a slightly more appropriate title than the one we chose).

Inhaltsverzeichnis

Frontmatter

Introduction

Introduction

Abstract
Every object described by a function f of a real variable gives rise to a “spectral” image described by the function
$$\hat{f}\left( \xi \right): = \frac{1}{{2\pi }}\int_{{ - \infty }}^{\infty } {f\left( t \right){{e}^{{ - it\xi }}}dt\quad \left( {\xi \in \mathbb{R}} \right).} $$
(1)
The right side makes sense for every ξ ∈ ℝ if fL1(ℝ). The function is called the Fourier transform of f. (Sometimes we write F(f) instead of ). The Fourier transform can be defined not only for a summable function but also for any tempered distribution.
Victor Havin, Burglind Jöricke

The Uncertainty Principle Without Complex Variables

Frontmatter

1. Functions and Charges with Semibounded Spectra

Abstract
Let us imagine a device transforming an input f into the output L(f); both are some functions of time (“signals”). The domain χ of the operator L (i.e., the set of all admissible inputs) and the range of L are vector spaces consisting of (generalized) functions defined on ℝ. We assume L to be linear. Set τ h (f)(t) := f(t - h). The signal τ h (f) is a shift of f; if h > 0, then τ h (f) is h time units later than f. Suppose that our device is indifferent to the choice of the origin of the time axis. In other words, τ h (χ) ⊂ χ, L(τ h (f)) = τ h L(f) (h ∈ ℝ, f ∈ χ). Such operators L are called shift invariant. It is well known that (under some natural restrictions) a shift invariant operator L can be represented as a convolution:
$$L\left( f \right) = a*f\quad \left( {f \in X} \right)$$
(1)
where a is a (generalized) function. It coincides with the output corresponding to the δ-input: a = L(δ). Real devices obey the causality principle (“no output without an input”). This means that if f ∈ χ, t0 ∈ ℝ and f | (-∞, t0) =0, then L(f) | (-∞, t0) = 0. At any given moment to such a device takes into account the past of the signal f and ignores its future; L(f) | (-∞, t0) is completely determined by f | (-∞, t0). It is easy to see that a device described by a shift invariant operator L is causal iff a | (-∞, 0) = 0.
Victor Havin, Burglind Jöricke

2. Some Topics Related to the Harmonic Analysis of Charges

Abstract
The theme of §§1–5 of this chapter can be worded as follows: “singular charges and vanishing of the Fourier transform at infinity”. These two properties of a charge seem to contradict each other and are difficult to reconcile. Indeed, if a non-zero charge μ ∈ M (T) is m-singular (i.e. if it belongs to M s ), then its support is “small”; but this fact, according to the Uncertainty Principle, is an obstacle for the spectral smallness of μ which is expressed by the condition
$$\mathop {\lim }\limits_{\left| n \right| \to + \infty } {\kern 1pt} \hat \mu \left( n \right) = 0$$
(1)
Nevertheless, the inclusion μ ∈ M s and (1) are compatible, and we describe some methods to construct non-zero m-singular r-charges (i.e. charges satisfying (1)). We use the last term in honor of Rajchman, who made one of the first contributions to our theme.
Victor Havin, Burglind Jöricke

3. Hilbert Space Methods

Abstract
Let us denote by X the real line ℝ or the unit circle 𝕋. Then will denote, respectively, ℝ or ℤ; L2(X) will mean the same as L2(X, m) where m is Lebesgue measure on ℝ or normalized Lebesgue measure on 𝕋. The symbol L2() denotes L2 (ℝ, m) or 2(ℤ).
Victor Havin, Burglind Jöricke

Complex Methods

Frontmatter

1. The Uncertainty Principle from the Complex Point of View. First Examples

Abstract
Classical Harmonic Analysis deals with distributions or functions defined on ℝ or on 𝕋. These sets are parts of ℂ, and analytic functions of a complex variable are always present (though not always visible) in Fourier analysis, however far from the analyticity the objects of analysis might be. The simplest explanation is this: partial Fourier sums ∑| n |≤N f̂(n)z n are rational functions and partial Fourier integrals are entire functions; both “live” in the whole of ℂ, not only in ℝ or 𝕋. Leaving 𝕋 and ℝ for ℂ we get a vast new perspective: various manifestations of the UP become uniqueness theorems which are the core of Complex Analysis. The following is a (rather primitive) example: if μ ∈ M(ℝ) and if diam supp μ < +∞, then μ̂ coincides on ℝ with an entire function; hence any charge whose support and spectrum are bounded is zero. Subtler uniqueness theorems yield far more precise and profound forms of the UP.
Victor Havin, Burglind Jöricke

2. The Logarithmic Integral Diverges

Abstract
In this chapter we discuss variants of the UP where the smallness of a function f means the divergence of its logarithmic integral L(f):
$$\mathcal{L}(f): = \left\{ {\begin{array}{*{20}{c}} {{{\smallint }_{\mathbb{T}}}\log |f|dm} & {\left( {f \in {{L}^{1}}(\mathbb{T},m)} \right),} \\ {{{\smallint }_{\mathbb{R}}}\log |f|d\Pi } & {\left( {f \in {{L}^{1}}(\mathbb{R},\Pi } \right).} \\ \end{array} } \right.$$
We denote by Π “the Poisson measure” (as in §1.5 of Part One): dΠ = = π-1(1 + x2)-1dx.
Victor Havin, Burglind Jöricke

3. The Logarithmic Integral Converges

Abstract
Divergence of a logarithmic integral is the principal sufficient condition of many forms of the UP. But in fact it coincides with the necessary condition. The logarithmic integral determines a border-line separating two realms, the one undividedly governed by the UP (and described in Chapter 2) and the other where the resistance to the UP is possible and non-zero pairs (f, f̂) are allowed whose elements are small simultaneously. This chapter is devoted to methods of construction (or at least to the existence proofs) of such pairs.
Victor Havin, Burglind Jöricke

4. Missing Frequencies and the Diameter of the Support. The Second Beurling-Malliavin Theorem and the Fabry Theorem

Abstract
The first five paragraphs of this chapter are devoted to the following form of the UP:
$$T\; \in \;\mathcal{D}'\left( \mathbb{R} \right),\quad \operatorname{supp} T \subset \left( { - a,a} \right),\;\hat{T}|\;\Lambda = 0 \Rightarrow \;T = 0 $$
where a is a positive number, Λ is a sufficiently “dense” set of real numbers (“frequencies”). The second Beurling-Malliavin theorem stated in §3 and proved in §§4–5 yields very precise conditions to be imposed onto the pair (a, Λ) to ensure this variant of the UP. The setting of the problem is discussed in 81; Sect. 5.8 contains some concrete examples.
Victor Havin, Burglind Jöricke

5. Local and Non-local Convolution Operators

Abstract
Let K be a linear operator mapping a linear set X ⊂ D’(ℝ d ) into D’(ℝ d ). We denote X and K(X) by dom K and im K (respectively). The operator K is called local if does not increase the support, i.e.
$$T\; \in \;dom\;k\; \Rightarrow \;\operatorname{supp} \;k\left( T \right)\; \subset \;\operatorname{supp} \;T $$
(Loc)
In other words, for any open O ⊂ℝd the distribution K (T)|O depends only on the restriction of T onto the same set O:
$$\${T_{1,}}{T_2}\; \in \;dom\;K,\;{T_1}|0 = {T_2}|0 \Rightarrow \;K\left( {{T_1}} \right)|0 = K\left( {{T_2}} \right)|0$$
(1)
.
Victor Havin, Burglind Jöricke

Backmatter

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