Description

Linear Operators
Assignment 4
Exercise 4.1
Calculate the Fourier-sine series of the function f defined on [0,π] and given by f(x) = x(π − x). Evaluate the series at a suitable point to find the value of the series
.
(4Marks)
Exercise 4.2
i) Show that
is an orthonormal system in (2Marks) ii) Show that cosk x, k ∈ N, is a linear combination of {1,cosx,cos(2x),…,cos(kx)}, so that
span{1,cosx,cos2 x,…,cosk x} = span{1,cosx,cos(2x),…,cos(kx)}.
(2Marks)
iii) Let ef ∈ C([0,π]). Consider the change of variables y = cosx, x ∈ [0,π], and define fe ∈ C([−1e,1]) by f(y) = f(arccosy). Use the Weierstraß Approximation Theorem to approximate uniformly f by polynomials, and show that this means that f can be approximated uniformly by finite linear combinations of 1,cosx,cos(2x),…. (2Marks)
iv) Conclude that spanB is dense in C([0,π]) in the ∥·∥ -norm.
∞
(2Marks)
v) Further deduce that spanB is dense in C([0,π]) in the ∥·∥L2-norm. (1Mark)
vi) Further deduce that spanB is dense in L2([0,π]) in the ∥·∥L2-norm (use that C([0,π]) is by definition dense in L2([0,π])). Hence, B is an orthonormal basis of L2([−π,π]). (1Mark)