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Exercise 2.4: Weighted inner products and approximation of singular func-
tions
Consider the function f (t) = 1/t in the interval (0, 1].
(a) Show that f(t) is not in L2(0, 1), but that it is in the Hilbert space L2, w (0, 1),
where the inner product is given by
(x, y)w =
x (t)y(t)w(t) dt
and w (t) = t2.
(b) From the set {1, t, to, to, t* ), construct a set of ON basis functions for 12, w (0, 1).
These are the first five Jacobi polynomials (Abramowitz and Stegun, 1970).
(c) Find a five-term approximation to 1/t with this inner product and basis. Plot
the exact function and five-term approximation. Compute the error between the
exact and approximate solutions using the inner product above to define a norm.
This type of inner product is sometimes used in problems where the solution
is known to show a singularity. As your analysis will show, polynomials can be
used to get a fairly good approximation except very near the singularity.

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