Reference no: EM13204640
Consider the singular Sturm-Liouville eigenvalue problem (d/dx(x du/dx))+ lambda x^-1 u=0
u(1) = 0:
a) Specify an appropriate boundary condition at x = 0 to ensure that the problem is self-adjoint.
b) Use the Rayleigh quotient to show that there are no negative eigenvalues.
c) Find two linearly independent solutions to the di erential equation.
d) Find the spectrum and eigenfunctions.
Problem 2
Suppose we want to approximate a piecewise continuous function f(x) on the interval by the trigonometric polynomial":
a) Determine the form of the coefficients that minimizes the total square deviation Integrate[(f(x)-Fn(x))^2),{x,-Pi,Pi}] of the approximation. Hint: Exploit orthogonality
b) After calculating optimal coecients for a given value of n, suppose we decide to increase n to obtain a better approximation with more terms. What modification is needed for the previously calculated coefficients?
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