The Fourier transformation of functions defined on a Hilbert space

The dimensional reduction method for solving boundary value problems of Helmholtz's equation in domain Ωd := ℝn × (-d,d) by replacing them with systems of equations in ℝn are investigated. Basic tool to analyze dimensional reduction technique for problems on an unbounded domain Ωd is the use of Fourier transformation. The error estimates between the exact solution and the dimensionally reduced solution in some Hilbert space are obtained when d and N are given. The rates of convergence depend on the smoothness of the data on the faces.

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DIMENSIONAL REDUCTION FOR THE BEAM IN ELASTICITY ON AN UNBOUNDED DOMAIN

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820259900021x ◽

1999 ◽

Vol 09 (03) ◽

pp. 415-444 ◽

Cited By ~ 1

Author(s):

KANG-MAN LIU

Keyword(s):

Hilbert Space ◽

Unbounded Domain ◽

Dimensional Reduction ◽

Fourier Transformation ◽

Reduction Method ◽

Rates Of Convergence ◽

Reduction Technique ◽

Systems Of Equations ◽

Basic Tool ◽

Dimensional Reduction Method

The dimensional reduction method is investigated for solving boundary value problems of the beam in elasticity on domain Ωd:=ℝ×(-d,d) by replacing the problems with systems of equations in ℝ. The basic tool to analyze the dimensional reduction technique for problems in an unbounded domain Ωd is using of Fourier transformation. The error estimates between the exact solution and the dimensionally reduced solution in a Hilbert space are obtained when d and N are given. The rates of convergence depend on the smoothness of the data on the faces.

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Simulations of high-resolution images of amorphous silicon films

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010014419x ◽

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.

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