Description
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.
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