Fourier transformation of generalized functions I

1987 ◽  
pp. 135-144
Keyword(s):  
Fourier Transformation ◽  
Generalized Functions

scholarly journals Algebraic and differential properties of polynomial Fourier transformation

2020 ◽  
Vol 53 (1) ◽  
pp. 59-68
Author(s):  
S.V. Sharyn
Keyword(s):  
Algebraic Structure ◽  
Fourier Transformation ◽  
Applied Mathematics ◽  
Adjoint Operator ◽  
Operator Method ◽  
Symmetric Tensor ◽  
Generalized Functions ◽  
Cauchy Problems ◽  
Distribution Spaces ◽  
Differential Properties

Methods of integral transformations of (generalized) functions are widely used in the solution of initial and boundary value problems for partial differential equations. However, many problems in applied mathematics require a nonlinear generalization of distribution spaces. Besides, an algebraic structure of a space of distributions is desirable, which is needed, for example, in quantum field theory.In the article, we use the adjoint operator method as well as technique of symmetric tensor products to extended the Fourier transformation onto the spaces of so-called polynomial rapidly decreasing test functions and polynomial tempered distributions. In such spaces it is possible to solve some Cauchy problems, for example, infinite dimensional heat equation associated with the Gross Laplacian.Algebraic and differential properties of the polynomial Fourier transformation are investigated. We prove some analogical to classical properties of this map. Unlike to the classic case, the spaces of polynomial test and generalized functions have algebraic structure. We prove that polynomial Fourier transformation acts as homomorphism of appropriate algebras. It is clear that the classical analogue of such property is absent.

scholarly journals The Spherical Tensor Gradient Operator

2005 ◽  
Vol 70 (8) ◽  
pp. 1225-1271 ◽  
Author(s):  
Ernst Joachim Weniger
Keyword(s):  
Fourier Transformation ◽  
Bessel Functions ◽  
Delta Function ◽  
Scalar Function ◽  
Addition Theorem ◽  
Generalized Functions ◽  
Addition Theorems ◽  
Gradient Operator ◽  
Multicenter Integrals ◽  
Gaussian Functions

The spherical tensor gradient operator Ylm(∇), which is obtained by replacing the Cartesian components of r by the Cartesian components of ∇ in the regular solid harmonic Ylm(r), is an irreducible spherical tensor of rank l. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank l. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of Ylm(∇) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. Math. London Soc. 24, 54 (1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if Ylm(∇) is applied to them. Fourier transformation is very helpful in understanding the properties of Ylm(∇) since it produces Ylm(-ip). It is also possible to apply Ylm(∇) to generalized functions, yielding for instance the spherical delta function δlm(r). The differential operator Ylm(∇) can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of rvYlm(r) with v ∈ πR.

The $ q$-Fourier transformation of $ q$-generalized functions

1999 ◽  
Vol 190 (5) ◽  
pp. 717-735 ◽  
Author(s):  
M A Ol'shanetskii ◽  
V-B K Rogov
Keyword(s):  
Fourier Transformation ◽  
Generalized Functions

Simulations of high-resolution images of amorphous silicon films

1986 ◽  
Vol 44 ◽  
pp. 544-545
Author(s):  
G. Y. Fan ◽  
J. M. Cowley
Keyword(s):  
Electron Microscope ◽  
High Frequency ◽  
Fourier Transformation ◽  
Amorphous Materials ◽  
Low Frequency ◽  
Chromatic Aberration ◽  
Structure Information ◽  
Frequency Components ◽  
High Resolution Images ◽  
Spatial Frequency Spectrum

It is well known that the structure information on the specimen is not always faithfully transferred through the electron microscope. Firstly, the spatial frequency spectrum is modulated by the transfer function (TF) at the focal plane. Secondly, the spectrum suffers high frequency cut-off by the aperture (or effectively damping terms such as chromatic aberration). While these do not have essential effect on imaging crystal periodicity as long as the low order Bragg spots are inside the aperture, although the contrast may be reversed, they may change the appearance of images of amorphous materials completely. Because the spectrum of amorphous materials is continuous, modulation of it emphasizes some components while weakening others. Especially the cut-off of high frequency components, which contribute to amorphous image just as strongly as low frequency components can have a fundamental effect. This can be illustrated through computer simulation. Imaging of a whitenoise object with an electron microscope without TF limitation gives Fig. 1a, which is obtained by Fourier transformation of a constant amplitude combined with random phases generated by computer.

Zur Anwendung der harmonischen Analyse bei der Funktionsszintigraphie des Herzens

1979 ◽  
Vol 18 (04) ◽  
pp. 167-171 ◽  
Author(s):  
R. Knopp ◽  
H.-P. Breuel ◽  
P. Fischer
Keyword(s):  
Fourier Transformation

ZusammenfassungDie nach Gleichverteilung von 99mTc-Serumalbumin szintigraphisch gewonnenen Volumenkurven des linken Ventrikels werden zur Gewinnung aussagefähiger Ergebnisse einer harmonischen Analyse unterzogen. Es ergibt sich hierdurch die Möglichkeit der quantitativen Analyse der Volumenkurve in geschlossener Form. Die Resynthese nach Fourier-Transformation stellt für den vorliegenden Funktionstyp ein gut geeignetes Glättungsverfahren dar, so daß die aus den Volumenkurven berechneten Parameter Ejektionsfraktion, maximale Austreibungs- und Füllungsgeschwindigkeit mit großer Genauigkeit erfaßt werden.

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