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Problem 1. Find a formula for the polynomial p of least degree that takes these values

p(xi) = yi

, p0

(xi) = 0 (0 ≤ i ≤ n)

Problem 2. Determine whether this is a quadratic spline function

f(x) =

x x ∈ (−∞, 1]

−1/2(2 − x)

2 + 3/2 x ∈ [1, 2]

3/2 x ∈ [2, ∞)

Problem 3. Determine all the values of a, b, c, d, e for which the following function is a cubic

spline function

f(x) =

a(x − 2)2 + b(x − 1)3 x ∈ (−∞, 1]

c(x − 2)2 x ∈ [1, 3]

d(x − 2)2 + e(x − 3)3 x ∈ [3, ∞)

Problem 4. Prove that an orthogonal set of nonzero functions is necessarily linearly independent.

Problem 5. Find a formula for dist(f, G), where G is the subspace spanned by an orthonormal

set of g1, g2, . . . , gn.

1

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