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186 CHAPTER 6 Vectors and Vector Spaces

2.5 2.5

2 2

1.5 1.5

1 1

0.5 0.5

–1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1
x x

FIGURE 6.17 f and f S in Example 6.24. FIGURE 6.18 f and f S ∗ in Example 6.24.

and
1
2 2
f − f S ∗ = ( f (x) − f S ∗) dx ≈ 0.000022.
−1
Then
f − f S ≈ 0.038 and f − f S ∗ ≈ 0.005.

SECTION 6.7 PROBLEMS

Problems 1 through 4, involve use of the Gram- The following problems are in the spirit of Example 6.24.
Schmidt orthogonalization process in a function space
2
5. Approximate f (x) = x on [0,π] with a linear combi-
C[a,b].
nation of the functions 1, cos(x),cos(2x),cos(3x),and
cos(4x).Use p(x)=1 in the weighted inner product on
1. In C[0,1], find an orthogonal set of two functions this function space. Graph f (x) and the approximating
that spans the same subspace as the two functions e −x linear combination on the same set of axes. Hint:Cal-
x
and e ,using p(x) = 1 in the weighted inner product culate f S , the orthogonal projection of f onto the sub-
integral. space of C[0,π] spanned by 1, cos(x),··· ,cos(4x).
2. In C[−π,π], find an orthogonal set of functions that 6. Repeat Problem 5, except now use the functions
spans the same subspace as sin(x),cos(x),and sin(2x). sin(x),··· ,sin(5x).
Use p(x) = 1 in the weighted inner product. 7. Approximate f (x)= x(2 − x) on [−2,2] using a linear
combination of the functions 1, cos(πx/2),cos(πx),
3. In C[0,1], find an orthogonal set of functions that spans
2
the same subspace as 1, x and x ,using p(x) = x in the cos(3πx/2),sin(πx/2),sin(πx),and sin(3πx/2).
Graph f and the approximating function on the same
weighted inner product.
set of axes. Hint:In C[−2,2],project f orthogonally
4. In C[0,2], find an orthogonal set of functions that spans onto the subspace spanned by the given functions. Use
the same subspace as 1,cos(πx/2),and sin(πx/2).Use the weight function p(x) = 1 in the inner product for
p(x) = x in the weighted inner product. this function space.

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