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Chapter 12: Q8-2P (page 562)

Find the norm of each of the following functions on the given interval and state the normalized functionP2(x)on(-1,1)

Short Answer

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Step by step solution

01

Concept of normalization factor

02

Calculate the norm of the function 

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Most popular questions from this chapter

Solve(22.9) to get(22.10) . If needed, see Chapter(8) , Section 2. The given equation(D+x)y0=0toobtainy0=e−x22.

Find the best (in the least squares sense) second-degree polynomial approximation to each of the given functions over the interval -1<x<1.

x4

Prove the least squares approximation property of Legendre polynomials [see (9.5) and (9.6)] as follows. Let f(x) be the given function to be approximated. Let the functions pl(x)be the normalized Legendre polynomials, that is, pl(x) = √(2l+1)/2 Pl(x) , so that

∫-11[pl(x)"]"2dx=1.

Show thatLegendre series for f(x)as far as the p2(x)term is

f(x)=c0p0(x)+c1p1(x) +c3p3(x) with c1 =∫-11f(x)pl(x) dx

Write the quadratic polynomial satisfying the least squares condition as b0p0(x)+b1p1(x)+b0p0(x)by Problem 5.14 any quadratic polynomial can be written in this form). The problem is to find b0, b1, b2so that I=∫-11[f2(x)+(b0-c0)2+(b1-c1)2+(b2-c2)2 -c02 -c12 -c22] dx

Now determine the values of the b's to make I as small as possible. (Hint: The smallest value the square of a real number can have is zero.) Generalize the proof to polynomials of degree n.

We obtained (19.10) forJp(x),p≥0.It is, however, valid forp≥-1, that is forNp(x),0≤p<1. The difficulty in the proof occurs just after (19.7); we said that u,v,u',v'are finite at x=0which is not true forNp(x).

However, the negative powers of x cancel ifp<1. Show this for p=12by using two terms of the power series (12.9) or (13.1) for the function N1/2(x)=-J-1/2(x) [see (13.3)].

Expand the following functions in Legendre series.

f(x) = P'n (x).

Hint: For I≥ n, ∫-11 P'n(x)Pl(x) dx=0 (Why?); for l<n, integrate by parts.
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