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Chapter 12: Q13P (page 582)
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
The second-degree polynomial approximation for the function f(x) = x4 is f(x)=1/5+5/7 (3x2-1).
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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.
Solve the differential equations in Problems 5 to 10 by the Frobenius method; observe that you get only one solution. (Note, also, that the two values of are equal or differ by an integer, and in the latter case the larger gives the one solution.) Show that the conditions of Fuchs's theorem are satisfied. Knowing that the second solution is X times the solution you have, plus another Frobenius series, find the second solution.
To study the approximations in the table, a computer plots on the same axes the given function together with its small approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agrees with the function for large . If the small approximation is not clear, plot it alone with the function over a small interval .
To show the first few terms of . Show that.
Find the best (in the least squares sense) second-degree polynomial approximation to each of the given functions over the interval -1<x<1.
cosπx
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