The theory of weak functions. I

1963 ◽  
Vol 276 (1365) ◽  
pp. 149-167 ◽  
Keyword(s):  
Continuous Function ◽  
Weak Convergence ◽  
Fourier Transforms ◽  
Generalized Functions ◽  
One Dimension ◽  
Test Functions ◽  
Weak Derivative

This paper develops the theory of distributions or generalized functions without any reference to test functions and with no appeal to topology, apart from the concept of weak convergence. In the calculus of weak functions, which is so obtained, a weak function is always a weak derivative of a numerical continuous function, and the fundamental techniques of multiplication, division and passage to a limit are considerably simplified. The theory is illustrated by application to Fourier transforms. The present paper is restricted to weak functions in one dimension. The extension to several dimensions will be published later.

scholarly journals A remark on convergence of test functions

1975 ◽  
Vol 20 (1) ◽  
pp. 73-76 ◽  
Author(s):  
W. F. Moss
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Generalized Functions ◽  
Test Functions ◽  
Definition Of ◽  
Theory Of Generalized Functions

In this note it is shown in the most frequently encountered spaces of test functions in the theory of generalized functions that the customary definitions of convergence are equivalent to apparently much weaker definitions. For example, in the space g the condition of uniform convergence of the functions together with all derivatives (which appears in the definition of convergence) is equivalent to the condition of pointwise convergence of the functions alone. Thus verification of convergence is simplified somewhat.

scholarly journals Almost periodic generalized functions

1983 ◽  
Vol 94 (1) ◽  
pp. 149-166
Author(s):  
H. Burkill ◽  
B. C. Rennie
Keyword(s):  
Generalized Functions ◽  
Inner Product ◽  
Almost Periodic ◽  
Test Functions ◽  
Complex Functions ◽  
Entire Complex ◽  
Order Of Magnitude ◽  
The Individual ◽  
Image Position ◽  
Derivatives Of

In (4) a space C of generalized functions was defined which is rather larger than the simple space used to such effect by Lighthill in (3). At the core of C is the space C0 = T of test functions. These are entire (complex) functions f such that all derivatives of f and its Fourier transform F have order of magnitude not exceeding as x → ± ∞, where c is a positive number depending on the individual derivative concerned. If f, g∈ T, the inner product 〈f | g〉 is defined to be

scholarly journals MEASUREMENT ERROR AND DECONVOLUTION IN SPACES OF GENERALIZED FUNCTIONS

2014 ◽  
Vol 30 (6) ◽  
pp. 1207-1246 ◽  
Author(s):  
Victoria Zinde-Walsh
Keyword(s):  
Measurement Error ◽  
Regression Models ◽  
Fourier Transforms ◽  
Generalized Functions ◽  
Well Posedness ◽  
Errors In Variables ◽  
Absolutely Continuous ◽  
Regression Functions ◽  
Deconvolution Problem ◽  
Convolution Equations

This paper considers convolution equations that arise from problems such as measurement error and nonparametric regression with errors in variables with independence conditions. The equations are examined in spaces of generalized functions to account for possible singularities; this makes it possible to consider densities for arbitrary and not only absolutely continuous distributions, and to operate with Fourier transforms for polynomially growing regression functions. Results are derived for identification and well-posedness in the topology of generalized functions for the deconvolution problem and for some regression models. Conditions for consistency of plug-in estimation for these models are provided.

scholarly journals Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

scholarly journals Stieltjes transforms on new generalized functions

2001 ◽  
Vol 25 (6) ◽  
pp. 421-427
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Analysis ◽  
Generalized Functions ◽  
Equivalence Classes ◽  
Generalized Function ◽  
Test Functions ◽  
Stieltjes Transform ◽  
Tempered Distributions ◽  
Analysis Techniques ◽  
Stieltjes Transforms

We introduce a Stieltjes transform on the equivalence classes of a new generalized function which has been successfully developed by Colombeau. Subsets of rapid descent test functions,𝒮(ℝn), as well as tempered distributions,𝒮′(ℝn), are used to preserve Fourier analysis techniques.

Introduction to Generalized Functions and Fourier Transforms

1975 ◽  
Vol 12 (2) ◽  
pp. 110-116
Author(s):  
V. Krishnan
Keyword(s):  
Fourier Analysis ◽  
Fourier Transforms ◽  
Slow Growth ◽  
Generalized Functions ◽  
Rapid Decay ◽  
Fourier Methods ◽  
Established Fact

Ever since the exposition of Schwartz on theory of distributions, the acceptance of Fourier methods for functions hitherto not amenable to rigorous Fourier analysis (e.g., impulse functions) has become an established fact. The introduction of the concepts of functions of ‘slow growth’ and ‘rapid decay’ provides a reinterpretation of the classical Fourier analysis.

Sign in / Sign up

Export Citation Format

Share Document