切比雪夫插值:互动之旅外文翻译资料

Chebyshev Interpolation: An Interactive Tour

Scott A. Sarra

Contents

1. Introduction

2. Chebyshev Polynomials

3. Continuous Chebyshev Expansion

4. Discrete Chebyshev Expansion

• 1. Interpolating Partial Sum
  2. Aliasing

5. Rates of Convergence

6. Filters

7. Current Research Areas

8. Further Explorations

9. Summary

10. References

________________________________________

1. Introduction

Most areas of numerical analysis, as well as many other areas of mathematics as a whole, make use of the Chebyshev polynomials. In several areas, e.g. polynomial approximation, numerical integration, and pseudospectral methods for partial differential equations, the Chebyshev polynomials take a significant role. In fact, the following quote has been attributed to a number of distinguished mathematicians:

'The Chebyshev polynomials are everywhere dense in numerical analysis.'

In this article we use Java applets to interactively explore some of the classical results on approximation using Chebyshev polynomials. We also discuss an active research area that uses the Chebyshev polynomials. Mason and Handscomb (2003) and Rivlin (1974) are devoted to the Chebyshev polynomials and may be consulted for more detailed information than we provide in this brief presentation. The Chebyshev polynomials are named for Pafnuty Chebyshev. You can read a brief biography of Chebyshev at Wikipedia

The article uses four applets:

bull; Chebyshev Polynomial (CP) applet

bull; Chebyshev Approximation (CA) applet

bull; Runge Phenomenon (RP) applet

bull; Exponential Filter (EF) applet

The CP applet and the CA applet are used frequently and thus open in separate windows that you can keep open as you read the text. The CA applet window also gives instructions for using the applet and definitions of the functions used in the applet.

2. Chebyshev Polynomials

The Chebyshev Polynomials (of the first kind) are defined by as

(1)

They are orthogonal with respect to the weight w(x)=(1-x²)^-1/2 on the interval [-1,1]. Intervals [a,b]other than[-1,1]are easily handled by the change of variables x. Although not immediately evident from definition (1), T_n is a polynomial of degree n. From definition (1) we have that T₀(x)==1 and T₁(x)=.=x

Exercise. Use basic trig identities to establish the triple recursion relation

(2)T_{n 1}(x)=2xT_n(x)-T_n-1(x), n=1, 2hellip;

Using equation (2) we see

and that the Chebyshev polynomial Tn is indeed a polynomial of degree n.

Applet Activity

What do the Chebyshev polynomials look like? The Chebyshev polynomials of degree n = 0, 1, hellip;, 12 can be plotted in the CP applet. Move the slider to change the degree. Notice that |T_n(x)|. Since Tn is a degree n polynomial we can observe as expected that it has n zeros, which in this case are real and distinct and located in[-1,1].

Exercise. Show that the zeros of Tn are

(3)x_k=, k=0,1,hellip;,n

The zeros are known as the Chebyshev-Gauss (CG) points.

3. Continuous Chebyshev Expansion

The infinite continuous Chebyshev series expansion is

(4)f(x)

where

(5) An=(1-x²)^-1/2Tn(x)dx

The single prime notation in the summation indicates that the first term is halved. Truncating the series after N 1 terms, we get the truncated continuous Chebyshev expansion:

(6)S_N(x)=

There are several functions in which the integral for the coefficients a_n can be evaluated explicitly, but this is not possible in general. Examples included in the CA applet for which a continuous truncated expansion can be derived are the sign function f₁, the square root function f₄, and the absolute value function f₅ (open the applet window to review the definitions of these functions).

The conditions which must be placed on f to ensure the convergence of the series (4) depend on the type of convergence to be established: pointwise, uniform, or L². At the lowest level, the series (4) converges pointwise to f at points where f is continuous in [-1,1] and converges to the left and right limiting values of f at any of a finite number of jump discontinuities in the interior of the interval.

Applet Activity

The sign function in the CA applet has a jump discontinuity at x0 = 0 and has the limiting values on each side of the discontinuity of f(x₀ )=1 and f(x₀^-)=-1 . Thus the series converges to zero at this point, i.e.

S_N(X₀)[f(x₀ ) f(x₀^-)]

for sufficiently large N. In the applet select the sign function from the Functions menu and check the blue continuous, S option on the Approximation menu. Using the slider at the bottom of the applet, slowly adjust N from N = 7 to N = 128 and observe that the value of S_N(0) is approximately zero.

Exercise. Show that if f is an even function then a_k=0 for k = 1, 3, 5, ... If f is an odd function then a_k=0 for k = 0, 2, 4, ....

Applet Activity

The result in the last exercise can be observed in the truncated continuous expansion of f₄(x)= and f₅(x)=|x| (even) and f₁(x) = sign(x) (odd) in the CA applet. For example, select the even function f4 which is labeled as sqrt on the Functions menu and select the blue continuous, S option on the Approximation menu. Then on the Options men

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译文

切比雪夫插值:互动之旅

2。切比雪夫多项式

这切比雪夫多项式(第一类)定义为

(1) T_n(x)=cos[n arcos(x)]

它们相对于重量是正交的w(x)=(1-x²)^-1/2 在幕间休息时[-1,1]。间隔[a,b]除了[-1,1]很容易通过变量的变化来处理x。虽然从定义上看不是很明显(1)，T_n是一个多项式n₀从定义(1)我们有那个T₀(x)==1和T₁(x)=.=x 。

锻炼。使用基本的触发器标识来建立三重递归关系

(2) T_{n 1}(x)=2xT_n(x)-T_n-1(x), n=1, 2hellip;

使用等式(2)我们明白了

T2(x)=2x(x)-1=2x^2-1

T3(x)=2x(2x^2-1)-x=4x^3-3x

T4(x)=2x(4x^3-3x)-(2x^2-1)=8x^4-8x^2 1

切比雪夫多项式Tn确实是一个多项式n。

小程序活动

切比雪夫多项式是什么样子的？度的切比雪夫多项式n= 0，1，...，12可以绘制在CP小程序。移动滑块以更改度数。请注意|T_n(x)| 。因为T_n是学位n多项式我们可以如预期的那样观察到n零，在这种情况下是真实的，不同的，位于[-1,1]。

例如。显示的T_n是零

(3) x_k=, k=0,1,hellip;,n

零被称为切比雪夫-高斯点数。

3。连续切比雪夫展开

这无限连续切比雪夫级数展开存在

(4) f(x)

在哪里

(5) An=(1-x²)^-1/2Tn(x)dx

求和中的单素数表示第一项减半。截断序列后N 1项，我们得到截断连续切比雪夫展开以下内容:

(6) S_N(x)=

系数的积分有几种函数 a_n可以显式评估，但这一般是不可能的。中包含的示例CA小程序符号函数是一个连续截断展开式f₁，平方根函数f₄，和绝对值函数f₅(打开applet窗口查看这些函数的定义)。

必须满足的条件f为了确保级数的收敛性(4)取决于要建立的收敛类型:逐点、一致或L²。在最低级别，系列(4)逐点收敛到f在f是连续的[-1,1] 并且收敛到左右极限值f在区间内部有限数量的跳跃间断处。

小程序活动

中的符号函数CA小程序在...处有跳跃中断x₀= 0，并且在间断的每一侧都有极限值f(x₀ )=1和f(x₀^-)=-1。因此，级数在这一点上收敛到零，即

S_N(X₀)[f(x₀ ) f(x₀^-)]

对于足够大的N。在小程序中，从功能菜单中选择符号功能，并在近似菜单中选中蓝色连续选项。使用小程序底部的滑块，慢慢调整N从N= 7至N= 128，并观察到 S_N(0)大约为零。

锻炼。显示如果f是一个偶数函数 a_k=为k= 1，3，5，...如果f是一个奇怪的函数 为k= 0，2，4，....

小程序活动

最后一个练习的结果可以在截断的连续扩展中观察到f₄(x)= 和f₅(x)=|x| (偶数)和f1（x)=符号(x)(奇数)在CA小程序。例如，选择偶数函数f4标记为sqrt在“功能”菜单上，选择蓝色连续的,S“近似”菜单上的选项。然后在选项菜单上检查绘图系数并使用滑块慢慢调整N从N= 7至N= 21。在右边的窗口观察到 a_k=0为k= 1，3，5，....系数的大小也可以通过y-对数缩放的轴(半科学在选项菜单上)。然而，在这种情况下，系数为零并未绘制成对数(0)未定义。

4。离散切比雪夫展开式

当积分在(5)无法精确计算，我们可以引入离散网格并使用数值求积(积分)公式。存在几种可能的网格和相关的求积公式。这切比雪夫-高斯-洛巴托(CGL)点

(7) x_k=-, k=0, 1,hellip;,N

是正交点的常见选择。CGL点是 n-1极值Tn(x) 出现加上时间间隔的端点[-1,1]。

小程序活动

使用CP小程序，观察切比雪夫多项式的极值是如何不均匀分布的，以及它们是如何围绕边界聚集的。在CA小程序，可以通过检查绘制CGL点绘制CGL点在“选项”菜单上。试着用符号函数从开始N= 9，然后随着增加N。

例如。展示一下Tn(x)=在n-1 CGL点。

相应的CGL求积公式是

(8)=

求和中的双素数表示第一项和最后一项减半。如果f是小于或等于的多项式2N-1，CGL求积公式是精确的。考虑到被积函数的值仅在N 1 CGL积分。使用CGL求积公式计算(5)，离散切比雪夫系数a_n 被定义为

(9) a_na_n=T_n(x_k)

还有离散截断部分和存在

(10) P_N(x)=a_n(x)

使用定义(9)拿走O(N²)浮点运算(flops)来评估离散切比雪夫系数。大的N，更好的选择是快速余弦变换(FCT)(布里格斯和汉森，1995年O(Nlog₂N)。例如，如果N = 1000,N²=1,000,000正在Nlog₂N10,000。FCT的极端效率是切比雪夫近似法在应用中流行的原因之一。

外文原文

2. Chebyshev Polynomials

The Chebyshev Polynomials (of the first kind) are defined by as

(1)T_n(x)=cos[n arcos(x)]

They are orthogonal with respect to the weight w(x)=(1-x²)^-1/2 on the interval [-1,1]. Intervals [a,b]other than[-1,1]are easily handled by the change of variables x. Although not immediately evident from definition (1), T_n is a polynomial of degree n. From definition (1) we have that T₀(x)==1 and T₁(x)=.=x

Exercise. Use basic trig identities to establish the triple recursion relation

(2)T_{n 1}(x)=2xT_n(x)-T_n-1(x), n=1, 2hellip;

Using equation (2) we see

T₂(x)=2x(x)-1=2x^2-1

T₃(x)=2x(2x^2-1)-x=4x^3-3x

T₄(x)=2x(4x^3-3x)-(2x^2-1)=8x^4-8x^2 1

and that the Chebyshev polynomial Tn is indeed a polynomial of degree n.

Applet Activity

What do the Chebyshev polynomials look like? The Chebyshev polynomials of degree n = 0, 1, hellip;, 12 can be plotted in the CP applet. Move the slider to change the degree. Notice that |T_n(x)|. Since Tn is a degree n polynomial we can observe as expected that it has n zeros, which in this case are real and distinct and located in[-1,1].

Exercise. Show that the zeros of Tn are

(3)x_k=, k=0,1,hellip;,n

The zeros are known as the Chebyshev-Gauss (CG) points.

3. Continuous Chebyshev Expansion

The infinite continuous Chebyshev series expansion is

(4)f(x)

where

(5) An=(1-x²)^-1/2Tn(x)dx

The single prime notation in the summation indicates that the first term is halved. Truncating the series after N 1 terms, we get the truncated continuous Chebyshev expansion:

(6)S_N(x)=

There are several functions in which the integral for the coefficients a_n can be evaluated explicitly, but this is not possible in general. Examples included in the CA applet for which a continuous truncated expansion can be derived are the sign function f₁, the square root function f₄, and the absolute value function f₅ (open the applet window to review the definitions of these functions).

The conditions which must be placed on f to ensure the convergence of the series (4) depend on the type of convergence to be established: pointwise, uniform, or L². At the lowest level, the series (4) converges pointwise to f at points where f is continuous in [-1,1] and converges to the left and right limiting values of f at any of a finite number of jump discontinuities in the interior of the interval.

Applet Activity

The sign function in the CA applet has a jump discontinuity at x0 = 0 and has the limiting values on each side of the discontinuity of f(x₀ )=1 and f(x₀^-)=-1 . Thus the series converges to zero at this point, i.e.

S_N(X₀

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