scholarly journals On the Integer Translates of a Compactly Supported Function: Dual Bases and Linear Projectors

1990 ◽  
Vol 21 (6) ◽  
pp. 1550-1562 ◽  
Author(s):  
Asher Ben-Artzi ◽  
Amos Ron
Keyword(s):  
Compactly Supported ◽  
Integer Translates ◽  
Compactly Supported Function

scholarly journals On The Convolution of a Box Spline With a Compactly Supported Distribution: Linear Independence for the Integer Translates

1991 ◽  
Vol 43 (1) ◽  
pp. 19-33 ◽  
Author(s):  
Charles K. Chui ◽  
Amos Ron
Keyword(s):  
Laplace Transform ◽  
Critical Points ◽  
Linear Independence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compactly Supported ◽  
Integer Translates ◽  
Box Spline ◽  
Finite Set ◽  
Necessary And Sufficient

AbstractThe problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

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

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document