Reconstruction of a compactly supported function from the discrete sampling of its Fourier transform

1999 ◽  
Vol 47 (12) ◽  
pp. 3356-3364 ◽  
Author(s):  
Jiahong Yin ◽  
A.R. De Pierro ◽  
Musheng Wei
Keyword(s):  
Fourier Transform ◽  
Discrete Sampling ◽  
Compactly Supported ◽  
Compactly Supported Function

Resolution enhancement of imaging small-scale portions in a compactly supported function

2010 ◽  
Vol 27 (2) ◽  
pp. 141 ◽  
Author(s):  
Hsin M. Shieh ◽  
Yu-Ching Hsu ◽  
Charles L. Byrne ◽  
Michael A. Fiddy
Keyword(s):  
Resolution Enhancement ◽  
Small Scale ◽  
Compactly Supported ◽  
Compactly Supported Function

scholarly journals ON THE NUMERICAL SIMULATION OF TIME-SPACE FRACTIONAL COUPLED NONLINEAR SCHRÖDINGER EQUATIONS UTILIZING WENDLAND’S COMPACTLY SUPPORTED FUNCTION COLLOCATION METHOD

2021 ◽  
Vol 26 (1) ◽  
pp. 94-115
Author(s):  
Bahar Karaman
Keyword(s):  
Fractional Derivatives ◽  
Von Neumann ◽  
Nonlinear Schrödinger ◽  
Compactly Supported ◽  
Time Space ◽  
Compactly Supported Function ◽  
Conformable Derivatives ◽  
The Stability ◽  
Compactly Supported Functions ◽  
Nicolson Scheme

This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, the time and space fractional derivatives are considered in terms of Caputo and Conformable derivatives, respectively. The present numerical discussion is based on the following ways: we first approximate the Caputo fractional derivative of the proposed equation by a scheme order O(∆t2−α), 0 < α < 1 and then the Crank-Nicolson scheme is employed in the mentioned equation to discretize the equations. Second, applying a linear difference scheme to avoid solving nonlinear systems. In this way, we have a linear, suitable calculation scheme. Then, the conformable fractional derivatives of the Wendland’s compactly supported functions are established for the scheme. The stability analysis of the suggested scheme is also examined in a similar way to the classic Von-Neumann technique for the governing equations. The efficiency and accuracy of the present method are verified by solving two examples.

OPERATOR-ADAPTED WAVELETS: CONNECTION WITH THE STRANG–FIX CONDITIONS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250006 ◽  
Author(s):  
VICTOR G. ZAKHAROV
Keyword(s):  
Differential Operators ◽  
Null Space ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Diagonal Form ◽  
Scaling Functions ◽  
Operator Matrix ◽  
Compactly Supported ◽  
Compactly Supported Function ◽  
Constant Coefficients

In this paper, we present an explicit method to construct directly in the x-domain compactly supported scaling functions corresponding to the wavelets adapted to a sum of differential operators with constant coefficients. Here the adaptation to an operator is taken to mean that the wavelets give a diagonal form of the operator matrix. We show that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions. We consider the necessary and sufficient conditions (actually the Strang–Fix conditions) on integer shifts of a compactly supported function (distribution) f ∈ S'(ℝ) to represent exactly any function from the null-space of a sum of differential operators with constant coefficients.

CONSTRUCTION OF SMOOTH COMPACTLY SUPPORTED WINDOWS GENERATING DUAL PAIRS OF GABOR FRAMES

2013 ◽  
Vol 06 (01) ◽  
pp. 1350011 ◽  
Author(s):  
Lasse Hjuler Christiansen ◽  
Ole Christensen
Keyword(s):  
Compact Support ◽  
Partition Of Unity ◽  
Dual Pair ◽  
New Method ◽  
Gabor Frames ◽  
Scaling Functions ◽  
Dual Pairs ◽  
Compactly Supported ◽  
Compactly Supported Function ◽  
New Construction

Let g be any real-valued, bounded and compactly supported function, whose integer-translates {Tkg}k∈ℤ form a partition of unity. Based on a new construction of dual windows associated with Gabor frames generated by g, we present a method to explicitly construct dual pairs of Gabor frames. This new method of construction is based on a family of polynomials which is closely related to the Daubechies polynomials, used in the construction of compactly supported wavelets. For any k ∈ ℕ ∪ {∞} we consider the Meyer scaling functions and use these to construct compactly supported windows g ∈ Ck(ℝ) associated with a family of smooth compactly supported dual windows [Formula: see text]. For any n ∈ ℕ the pair of dual windows g, hn ∈ Ck(ℝ) have compact support in the interval [-2/3, 2/3] and share the property of being constant on half the length of their support. We therefore obtain arbitrary smoothness of the dual pair of windows g, hn without increasing their support.

Measures with Fourier Transforms in L2 of a Half-space

2011 ◽  
Vol 54 (1) ◽  
pp. 172-179
Author(s):  
Bassam Shayya
Keyword(s):  
Fourier Transform ◽  
Lebesgue Measure ◽  
Half Space ◽  
Fourier Transforms ◽  
Geometric Information ◽  
Absolutely Continuous ◽  
Compactly Supported ◽  
The Fourier Transform

AbstractWe prove that if the Fourier transform of a compactly supported measure is in L2 of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

scholarly journals Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hiroshi Fujiwara ◽  
Kamran Sadiq ◽  
Alexandru Tamasan
Keyword(s):  
Convex Hull ◽  
Hilbert Transform ◽  
Analytic Functions ◽  
Two Dimensions ◽  
X Ray ◽  
Ray Transform ◽  
Compactly Supported ◽  
Compactly Supported Function ◽  
The Hilbert Transform ◽  
Partial Inversion

<p style='text-indent:20px;'>In two dimensions, we consider the problem of inversion of the attenuated <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula>-analytic functions in the sense of Bukhgeim.</p>

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