#### 2.6.8 So­lu­tion herm-h

Ques­tion:

A com­plete set of or­tho­nor­mal eigen­func­tions of on the in­ter­val 0 that are zero at the end points is the in­fi­nite set of func­tions

Check that these func­tions are in­deed zero at 0 and , that they are in­deed or­tho­nor­mal, and that they are eigen­func­tions of with the pos­i­tive real eigen­val­ues

Com­plete­ness is a much more dif­fi­cult thing to prove, but they are. The com­plete­ness proof in the notes cov­ers this case.

An­swer:

Any eigen­func­tion of the above list can be writ­ten in the generic form where is a pos­i­tive whole num­ber, in other words where is a nat­ural num­ber (one of 1, 2, 3, 4, ....) If you show that the stated prop­er­ties are true for this generic form, it means that they are true for every eigen­func­tion.

First check the end points. The graph of the sine func­tion, [1, item 12.22], shows that a sine is zero when­ever its ar­gu­ment is a whole mul­ti­ple of . That makes both and zero. So must be zero at 0 and too.

Now check that the norm of the eigen­func­tions is one. First find the norm of by it­self:

Since the sine is real, the com­plex con­ju­gate does not do any­thing, and you get

us­ing [1, item 18.26]. Di­vid­ing this by , the norm be­comes one; every eigen­func­tion is nor­mal­ized.

To ver­ify that is or­thog­o­nal to every other eigen­func­tion, take the generic other eigen­func­tion to be with a nat­ural num­ber dif­fer­ent from . You must then show that the in­ner prod­uct of these two eigen­func­tions is zero. Since the nor­mal­iza­tion con­stants do not make any dif­fer­ence here, you can just show that is zero. You get

us­ing again [1, item 18.26].

Ahem. Com­plete­ness. Well, just don't worry about it. There are a heck of a lot of func­tions here. In­fi­nitely many of them, to be pre­cise. Surely, with in­fi­nitely many func­tions, you should be able to ap­prox­i­mate any given func­tion to good ac­cu­racy?

(This state­ment is, of course, de­lib­er­ately lu­di­crous. In fact, if you leave out a sin­gle eigen­func­tion, say the func­tion, the re­main­ing in­fi­nitely many func­tions , , ...can sim­ply not re­pro­duce it by them­selves. The best they can do is be­ing zero and not try to ap­prox­i­mate at all. Still, if you do in­clude in the se­quence, any (rea­son­able) func­tion can be de­scribed ac­cu­rately by a com­bi­na­tion of the sines. It was hard to prove ini­tially; in fact, Fourier in his the­sis did not. The first proof is due to Dirich­let.)