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The theory of generalized functions is a general method that makes it possible to consider and compute divergent integrals, sum divergent series, differentiate discontinuous functions, perform the operation of integration to any complex power and carry out other such operations that are impossible in classical analysis. Such operations are widely used in mathematical physics and the theory of differential equations, where the ideas of generalized func­ tions first arose, in other areas of analysis and beyond. The point of departure for this theory is to regard a function not as a mapping of point sets, but as a linear functional defined on smooth densi­ ties. This route leads to the loss of the concept of the value of function at a point, and also the possibility of multiplying functions, but it makes it pos­ sible to perform differentiation an unlimited number of times. The space of generalized functions of finite order is the minimal extension of the space of continuous functions in which coordinate differentiations are defined every­ where. In this sense the theory of generalized functions is a development of all of classical analysis, in particular harmonic analysis, and is to some extent the perfection of it. The more general theories of ultradistributions or gener­ alized functions of infinite order make it possible to consider infinite series of generalized derivatives of continuous functions.

Inhaltsverzeichnis

Frontmatter

Distributions and Harmonic Analysis

Frontmatter

Introduction

Abstract
The theory of generalized functions is a general method that makes it possible to consider and compute divergent integrals, sum divergent series, differentiate discontinuous functions, perform the operation of integration to any complex power and carry out other such operations that are impossible in classical analysis. Such operations are widely used in mathematical physics and the theory of differential equations, where the ideas of generalized functions first arose, in other areas of analysis and beyond.
V. P. Havin, N. K. Nikol’skij

Chapter 1. The Elementary Theory

Abstract
The Space of Test Functions D = D(R1. The space D = D(R1 consists of the infinitely differentiable complex-valued functions of compact support on the line. The functions ϕ is said to have compact support if there exists a number Rsuch that \( \varphi (x) = 0if|x| > R. \) The set D is a vector space over the field C of complex numbers.
V. P. Havin, N. K. Nikol’skij

Chapter 2. The General Theory

Abstract
Densities and Generalized Functions on a Manifold. Let X be a smooth manifold, i.e., a manifold of class C. A density bundle on X is a onedimensional complex bundleIIx constructed as follows. Suppose an atlas is distinguished on X sonsisting of charts
$$ {\phi _a}:{X_a} \to {\mathbb{R}^n},n = n(a),a \in A, $$
(3.1.1)
where {Xa} is an open covering of X and let
$$ {\phi _{\beta a}}:{\phi _a}({X_a} \cap {X_\beta }) \to {\phi _\beta }({X_\beta } \cap {X_a}) $$
be the transition mappings of these charts.
V. P. Havin, N. K. Nikol’skij

Chapter 3. The Fourier Transform

Abstract
The Invariant Form of the Transform. We write the classical Fourier integral as 2
$$ F(p) = g(\xi ) \equiv \int\limits_x {e^{ - 2\pi i\xi x} f(x)dx,X} = \mathbb{R}^n . $$
(3.1.1)
(3.1.1)
V. P. Havin, N. K. Nikol’skij

Chapter 4. Special Problems

Abstract
Let f: XY be a mapping of smooth manifolds. The point x εX is said to be critical for f, and f(x) is a critical value of / if rang df(x) < dimf(x)Y, In particular an imbedding X ↩ Y is critical at all points x ε X where the dimension of X is less than the dimension of Y. In §3 of Chapt. 2, we defined the inverse image of any generalized function defined on Y under a submersion f: XY, i.e., under a mapping that has no critical points.
V. P. Havin, N. K. Nikol’skij

Chapter 5. Contact Structures and Distributions

Abstract
The most important source of the theory of generalized functions has been problems that arise in the theory of hyperbolic differential equations (including analytic questions of field theory). In particular the generalized functions that we call Hadamard kernels (Chapt. 1) were introduced under the name “nonsingular integrals of a new type” to construct a fundamental solution of a hyperbolic equation (Hadamard 1932). The work of Sobolev (1936) is devoted to the solution of this same problem. In the paper of M. Riesz (1949) the important idea of analytic continuation of a family of generalized functions arose in the same context. A new class of singular functions arises in the analysis of a wave field in a neighborhood of a caustic. Among the pioneering works in this area is the paper of V.M. Babich (1961a), where new special functions are introduced to study a field near an elementary caustic.
V. P. Havin, N. K. Nikol’skij

Optical and Acoustic Fourier Processors

Frontmatter

§1. Introduction

Abstract
A Fourier processor is a physical device inside which the Fourier image of a signal falling on the entrance to the device is formed. Signals processed in such devices are usually either radio-electronic signals propagating in electronic (radio) circuits, or light beams arriving through optical devices. Along with the electronic and optical elements in Fourier processors widespread use is made of acoustic components. In many Fourier processors repeated transformation of electronic, optical and acoustic signals into one another occurs. The central elements by means of which the Fourier transform is realized are usually the optical and acoustic components of the processors. The use of such devices has become widespread. In this connection a special area in optics has even arisen, known as Fourier optics. Optical and acoustic processors are used as computer parts or specialized devices for complex signal processing. Like the majority of computing devices of this kind, they have both natural advantages and natural disadvantages in comparison with purely digital computers. The sphere of application of Fourier processors in which the decisive value lies with their advantages is rapidly spreading.
V. P. Havin, N. K. Nikol’skij

§2. The Optical Fourier Transform

Abstract
In an optical Fourier processor the operation of convolution is realized in the propagation of a rectilinear light beam in the homogeneous space between two planes orthogonal to the beam. The operation of multiplication by a suitable function is realized by interrupting the beam en route with a thin lens.
V. P. Havin, N. K. Nikol’skij

§3. Notes and Comments

Abstract
The Influence of the Lens Size. Of the parameters of the various components of an optical system, by means of which the Fourier transform is carried out, the one having most influence on the boundedness of the section of the beam is the size of the lens. Up to now we have neglected the effects caused by the bounded dimensions of the lens. Let us now consider the question of these effects in the context of Scheme 1 (Fig. 4).
V. P. Havin, N. K. Nikol’skij

§4. Acoustic and Acousto-Optical Fourier Processors

Abstract
A variety of different analog devices are used to obtain the Fourier image of a radio-electronic signal. Such devices, however, are almost never constructed of purely electronic elements. Acoustic components, specifically crystals in which elastic waves, i.e., (ultra)-sonic waves, can propagate at the frequency of the radio signals, are often introduced to build them into a system of electronic elements. In doing this one chooses crystals possessing piezoelectric properties. Piezoelectric crystals are characterized by the fact that the propagation of elastic waves in them is accompanied by the appearance of an electric field. The electric field in turn causes elastic deformations in piezoelectrics. The use of acoustic components relies mainly on the fact that the speed of sound is much less than the speed of propagation of an electronic signal. As a consequence radio-electronic components that would otherwise have to be tens of meters in size can be replaced by acousto-electronic elements only centimeters in size.
V. P. Havin, N. K. Nikol’skij

The Uncertainty Principle in Harmonic Analysis

Frontmatter

Introduction

Abstract
The Fourier transform assigns to a function x defined on the real line R its “spectral representation” x:
$$ \widehat x(\xi ) = \frac{1}{{2\pi }}\int\limits_\mathbb{R} {x(t){e^{ - it\xi }}dt()\xi \in } \mathbb{R}). $$
V. P. Havin, N. K. Nikol’skij

Chapter 1. The Uncertainty Principle Without Complex Variables

Abstract
The most profound and delicate results relating to the uncertainty principle have been obtained using complex analysis. However we shall begin by talking about versions of the uncertainty principle established by purely real-variable methods.
V. P. Havin, N. K. Nikol’skij

Chapter 2. Complex Methods

Abstract
The complex point of view opens wide prospects for our theme: in its different manifestations the uncertainty principle arises as a corollary of the various uniqueness theorems in which complex analysis is so rich. Here is a crude example: the fact that any charge μ ε M (ℝ) with bounded support and bounded spectrum vanishes identically follows immediately from the fact that µ coincides on ℝ with some entire function if diam supp µ < +∞. It is clear that this primitive consideration can be developed into much more precise results.
V. P. Havin, N. K. Nikol’skij

Backmatter

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