特征算子谱表示与特征展开的研究

doi:10.11959/j.issn.1000-436x.2020096

通信学报 ›› 2020, Vol. 41 ›› Issue (5): 1-8.doi: 10.11959/j.issn.1000-436x.2020096

• 学术论文 • 下一篇

特征算子谱表示与特征展开的研究

王宏禹,邱天爽()

大连理工大学电子信息与电气工程学部，辽宁大连 116024

修回日期:2020-03-27 出版日期:2020-05-25 发布日期:2020-05-30
作者简介:王宏禹（1929- ），男，北京人，大连理工大学教授、博士生导师，主要研究方向为随机与时变信号处理|邱天爽（1954- ），男，江苏海门人，博士，大连理工大学教授、博士生导师，主要研究方向为射频与医学信号处理
基金资助:
国家自然科学基金资助项目(61671105);国家自然科学基金资助项目(61172108);国家自然科学基金资助项目(61139001)

Study on spectral representation and eigenexpansion based on eigen-operators

Hongyu WANG,Tianshuang QIU()

Faculty of Electronic Information and Electrical Engineering,Dalian University of Technology,Dalian 116024,China

Revised:2020-03-27 Online:2020-05-25 Published:2020-05-30
Supported by:
The National Natural Science Foundation of China(61671105);The National Natural Science Foundation of China(61172108);The National Natural Science Foundation of China(61139001)

摘要/Abstract

摘要：

深入研究了特征算子的谱表示与特征展开。给出了特征微分方程格林函数与厄尔密特微分算子及厄尔密特积分算子的关系式，以及厄尔密特微分算子与厄尔密特积分算子两者互逆的关系式；给出了厄尔密特微分算子的谱表示，指出有限区间斯-刘特征方程不能用于实现无穷维的谱表示式，厄尔密特微分算子的谱表示比诺伊曼研究简单清楚得多，具有优越性；给出了厄尔密特积分算子的特征展开（特征分解），具有理论一般性与全面性的优点，对文献[2]中将其用于研究特征谱表示的不正确论述进行了更正；给出了最优特征展开中长球面波函数命名的物理与几何意义。

关键词: 格林函数, 斯-刘微分方程, 特征算子, 谱表示, 特征展开, 长球面波函数

Abstract:

The spectral representation and expansion based on eigen-operators were deeply studied.The relations between Green’s function of characteristic differential equation and Hermitian differential and integral operators were given.The inverse relations between Hermitian differential operator and Hermitian integral operator were also studied.The spectral representation of Hermitian differential operators was given.It was also shown that the S-L eigen-equation cannot be used to realize the spectral representations of infinite dimensions in a finite interval.The method is much simpler and clearer than that of Neumann,and has advantages.The eigen-expansion (eigen-decomposition) of the Hermitian integral operator was given,which has the advantages of theoretical generality and comprehensiveness.The incorrect discussion in Wang et al<sup>[2]</sup>was correct that it is used to study the representation of characteristic spectrum.The physical and geometric meanings of naming for the long spherical wave function in optimal eigen-expansion were given.

Key words: Green’s function, S-L differential equation, eigen-operator, spectral representation, eigen-expansion, long spherical wave function

中图分类号:

TN911.7

王宏禹,邱天爽. 特征算子谱表示与特征展开的研究[J]. 通信学报, 2020, 41(5): 1-8.

Hongyu WANG,Tianshuang QIU. Study on spectral representation and eigenexpansion based on eigen-operators[J]. Journal on Communications, 2020, 41(5): 1-8.

图/表 4

表1

加权斯-刘微分方程边值情况"

微分方程	区间	边界条件	特征值	标准正交函数系
调和微分方程
$p (x) = 1, q (x) = 0, ρ (x) = 1$	(0,l)	y (0)=0	${(\frac{n π}{l})}^{2}$	$\sqrt{\frac{2}{l}} \sin \frac{n π}{l} x$
$\frac{d^{2} y}{d x^{2}} + λ y = 0$		y(l)=0	n=1,2,3,"
勒让德微分方程
$p (x) = (1 - x^{2}), q (x) = 0, ρ (x) = 1$	(-1,1)	y(±1)有界	n (n+1)	$\sqrt{n + \frac{1}{2}} p_{n} (x)$
$\frac{d}{d x} [(1 - x^{2}) \frac{d y}{d x}] + λ y = 0$			n=0,1,2,"
奇点x=±1
连带勒让德微分方程
$p (x) = (1 - x^{2}), q (x) = \frac{m^{2}}{1 - x^{2}}, ρ (x) = 1$	(-1,1)	y(±1)有界	n (n+1)	$\sqrt{n + \frac{1}{2}} p_{n} (x)$
$\frac{d}{d x} [(1 - x^{2}) \frac{d y}{d x}] + (λ - \frac{m^{2}}{1 - x^{2}}) y = 0$			n=0,1,2,"
奇点x=±1
贝塞尔微分方程
$p (r) = r, q (r) = \frac{n^{2}}{r}, ρ (r) = r$	(0,l)	y (0)有界	$k_{i}^{n}$	$\frac{\sqrt{2} r}{l J_{n + 1} (k_{i}^{n} l)} J_{n} (k_{i}^{n} r)$
$\frac{d}{d r} (r \frac{d y}{d r}) + (k^{2} r - \frac{n^{2}}{r}) y = 0$		y (kl)=0	i= 1,2,3,"
奇点r=0
埃尔米特微分方程
$p (x) = e^{- x^{2}}, q (x) = e^{- x^{2}}, ρ (x) = e^{- x^{2}},$	(-∞,∞)	y (±∞)与∞的有限次密同阶	2n+1,	$\frac{1}{\sqrt{2^{n} n! \sqrt{π}}} e^{- \frac{x^{2}}{2}} H_{n} (x)$
$\frac{d}{d x} (e^{- x^{2}} \frac{d}{d y}) + (\frac{λ}{α} e^{- x^{2}} - e^{- x^{2}}) y = 0$			n=0,1,2,"

表1

图1

图2

图3

参考文献 10

[1]	王宏禹, 邱天爽 . 微分方程格林函数法的研究[J]. 信号处理, 2015,31(4): 379-385.
	WANG H Y , QIU T S . Study on the differential equation using the Green’s function[J]. Signal Processing, 2015,31(4): 379-385.
[2]	王宏禹, 邱天爽 . 信号特征谱表示与平稳随机信号谱分解统一的研究[J]. 通信学报, 2018,39(12): 1-9.
	WANG H Y , QIU T S . Unified study on characteristic spectrum representation of signals and spectral decomposition for stationary random signals[J]. Journal on Communications, 2018,39(12): 1-9.
[3]	王宏禹, 邱天爽 . 零阶角长球面波函数与长球面波函数的研究[J]. 信号处理, 2013,29(7): 790-799.
	WANG H Y , QIU T S . Zero-order angular prolate spheroidal wave and prolate spheroidal wave functions[J]. Signal Processing, 2013,29(7): 790-799.
[4]	STEIN S . Addition theorems for spherical wave functions[J]. Quarterly of Applied Mathematics, 1961,19(1): 15-24.
[5]	MUKHTAROV O S , KADAKAL M , MUHTAROV F ? . Eigenvalues and normalized eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions[J]. Reports on Mathematical Physics, 2004,54(1): 41-56.
[6]	王宏禹 . 信号处理相关理论综合与统一法[M]. 北京: 国防工业出版社, 2005.
	WANG H Y . Synthesis and unification method in correlation theory of signal processing[M]. Beijing: National Defense Industry PressPress, 2005.
[7]	ECONOMOU E N . Green’s functions in quantum physics[M]. New York: SpringerPress, 1983.
[8]	王宏禹, 邱天爽 . 非平稳确定性信号与非平稳随机信号统一分类法的探讨[J]. 通信学报, 2015,36(2): 1-10.
	WANG H Y , QIU T S . Unified classification methods for determinate nonstationary signals and random nonstationary signals[J]. Journal on Communications, 2015,36(2): 1-10.
[9]	BADER R F W , GATTI C . A Green’s function for the density[J]. Chemical Physics Letters, 1998,287(3-4): 233-238.
[10]	王宏禹, 邱天爽 . 确定性信号与平稳随机信号分解的统一研究[J]. 通信学报, 2016,37(10): 1-8.
	WANG H Y , QIU T S . Unified study on the decomposition for deterministic signals and stationary random signals[J]. Journal on Communications, 2016,37(10): 1-8.

特征算子谱表示与特征展开的研究

Study on spectral representation and eigenexpansion based on eigen-operators

在线阅读

PDF下载

可视化

摘要/Abstract

引用本文

使用本文

图/表 4

参考文献 10

相关文章 1

Metrics

推荐阅读 0