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11.
In this paper κ-order slant Toeplitz operator on the Bergman space is defined. Some properties like spectrum, commuting are discussed. 相似文献
12.
一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度. 相似文献
13.
Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions() of the process is known, we prove that ifs() is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.Research supported in part by HKRGC grant no. 221600070, ONR contract no. N00014-90-J-1695 and DOE grant no. DE-FG03-87ER25037. 相似文献
14.
For the zero-temperature Glauber dynamics of theq-state Potts model, the fractionr(q, t) of spins which never flip up to timet decays like a power lawr(q, t)t
–(q)
when the initial condition is random. By mapping the problem onto an exactly soluble one-species coagulation model (A+AA) or alternatively by transforming the problem into a free-fermion model, we obtain the exact expression of (q) for all values ofq. The exponent (q) is in general irrational, (3)=0.53795082..., (4)=0.63151575..., ..., with the exception ofq=2 andq=, for which (2)=3/8 and ()=1. 相似文献
15.
Stefano Serra 《BIT Numerical Mathematics》1996,36(1):135-142
In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue
1
(n)
of symmetric (Hermitian)n ×n Toeplitz matricesT
n
(f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that
1
(n)
tends to essinff =m
f
in the following way:
1
(n)
–m
f
1/n
2k
. These authors use three assumptions:A1)f –m
f
has a zero inx =x
0 of order 2k.A2)f is continuous and at leastC
2k
in a neighborhood ofx
0.A3)x =x
0 is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L
1[–, ]. In this paper we further extend this theory to the case of a functionf L
1[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m
f
is the only parameter which characterizes the rate of convergence of
1
(n)
tom
f
. 相似文献
16.
Xiao-Qing Jin 《BIT Numerical Mathematics》1996,36(1):101-109
We study methods for solving the constrained and weighted least squares problem min
x
by the preconditioned conjugate gradient (PCG) method. HereW = diag (1, ,
m
) with 1
m
0, andA
T
= [T
1
T
, ,T
k
T
] with Toeplitz blocksT
l
R
n × n
,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A
T
= 0, whereM =W
–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E. 相似文献
17.
A good preconditioner is extremely important in order for the conjugate gradients method to converge quickly. In the case of Toeplitz matrices, a number of recent studies were made to relate approximation of functions to good preconditioners. In this paper, we present a new result relating the quality of the Toeplitz preconditionerC for the Toeplitz matrixT to the Chebyshev norm (f– g)/f, wheref and g are the generating functions forT andC, respectively. In particular, the construction of band-Toeplitz preconditioners becomes a linear minimax approximation problem. The case whenf has zeros (but is nonnegative) is especially interesting and the corresponding approximation problem becomes constrained. We show how the Remez algorithm can be modified to handle the constraints. Numerical experiments confirming the theoretical results are presented. 相似文献
18.
《Indagationes Mathematicae》2017,28(1):91-107
Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system. 相似文献
19.
This paper presents an infrared image super-resolution method based on compressed sensing (CS). First, the reconstruction model under the CS framework is established and a Toeplitz matrix is selected as the sensing matrix. Compared with traditional learning-based methods, the proposed method uses a set of sub-dictionaries instead of two coupled dictionaries to recover high resolution (HR) images. And Toeplitz sensing matrix allows the proposed method time-efficient. Second, all training samples are divided into several feature spaces by using the proposed adaptive k-means classification method, which is more accurate than the standard k-means method. On the basis of this approach, a complex nonlinear mapping from the HR space to low resolution (LR) space can be converted into several compact linear mappings. Finally, the relationships between HR and LR image patches can be obtained by multi-sub-dictionaries and HR infrared images are reconstructed by the input LR images and multi-sub-dictionaries. The experimental results show that the proposed method is quantitatively and qualitatively more effective than other state-of-the-art methods. 相似文献
20.
Fabrizio Colombo Samuele Mongodi Marco Peloso Stefano Pinton 《Mathematical Methods in the Applied Sciences》2019,42(5):1662-1686
Let e?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory. 相似文献