286 CHAPTER 11. INNER PRODUCT SPACES

12. A regular Sturm Liouville problem involves the differential equation, for an un-known function of x which is denoted here by y,

(p (x) y′)′+ (λq (x) + r (x)) y = 0, x ∈ [a, b]

and it is assumed that p (t) , q (t) > 0 for any t ∈ [a, b] and also there are boundaryconditions,

C1y (a) + C2y′ (a) = 0

C3y (b) + C4y′ (b) = 0

whereC2

1 + C22 > 0, and C2

3 + C24 > 0.

There is an immense theory connected to these important problems. The constant, λis called an eigenvalue. Show that if y is a solution to the above problem correspondingto λ = λ1 and if z is a solution corresponding to λ = λ2 ̸= λ1, then∫ b

a

q (x) y (x) z (x) dx = 0. (11.9)

and this defines an inner product. Hint: Do something like this:

(p (x) y′)′z + (λ1q (x) + r (x)) yz = 0,

(p (x) z′)′y + (λ2q (x) + r (x)) zy = 0.

Now subtract and either use integration by parts or show

(p (x) y′)′z − (p (x) z′)

′y = ((p (x) y′) z − (p (x) z′) y)

′

and then integrate. Use the boundary conditions to show that y′ (a) z (a)−z′ (a) y (a) =0 and y′ (b) z (b)−z′ (b) y (b) = 0. The formula, 11.9 is called an orthogonality relation.It turns out there are typically infinitely many eigenvalues and it is interesting to writegiven functions as an infinite series of these “eigenfunctions”.

13. Consider the continuous functions defined on [0, π] , C ([0, π]) . Show (f, g) ≡∫ π

0fgdx

is an inner product on this vector space. Show the functions{√

2π sin (nx)

}∞

n=1are

an orthonormal set. What does this mean about the dimension of the vector space

C ([0, π])? Now let VN = span(√

2π sin (x) , · · · ,

√2π sin (Nx)

). For f ∈ C ([0, π]) find

a formula for the vector in VN which is closest to f with respect to the norm determinedfrom the above inner product. This is called the N th partial sum of the Fourier seriesof f . An important problem is to determine whether and in what way this Fourierseries converges to the function f . The norm which comes from this inner product issometimes called the mean square norm.

14. Consider the subspace V ≡ ker (A) where

A =

1 4 −1 −1

2 1 2 3

4 9 0 1

5 6 3 4

Find an orthonormal basis for V. Hint: You might first find a basis and then use theGram Schmidt procedure.