Question 1. (15 marks) d2
(a) Show that —@ is a hermitian linear transformation, with respect to the inner product
L
(La) = 0: f0 f<x)*g(x) dx (0: e Rm),
when acting on the vector space of (nice) functions satisfying f ’(0) = f ’(L) = 0. (2 marks) (b) Use this to explain how to justify the following eigenfunction expansion: 1 00
f(x) = an + 2a,, cos g.
71:1 [You need not justify the actual form of the eigenfunctions] (2 marks)
(c) Deduce the Fourier-theoretic formulae for the an, ﬁxing a in the process. (3 marks) (d) Thereby determine the eigenfunction expansion of f(x)= (1— x)2, O < x < 1. (5 marks) (e) Determine the function that this eigenfunction expansion converges to, explaining your answer by quoting an
appropriate result from the slides. Sketch its graph for —2 s x s 2. Is the convergence uniform? (3 marks)