Using Exercises 5.10.5 and 5.10.6, prove that n=1 n2 n = pf df dx b a + b a p df dx2 qf2 dx. (5.10.15) [Hint: Let g = L(f), assuming that term-by-term differentiation is justified.] Furthermore, Chapter 9 introduces a Greens function G(x, x0), which is shown to satisfy n=1 2 n n = G(x, x). (5.10.17) Using (5.10.15), (5.10.16), and (5.10.17), derive an upper bound for the pointwise error (in cases in which the generalized Fourier series is pointwise convergent). Examples and further discussion of this are given by Weinberger [1995].

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