# Practice Final

This exam is closed book, no books, papers or recording devices permitted. You may use theorems from class, or the book, provided you can recall them correctly.

Problem 1

Suppose and for all simple measurable functions on Show thatt almost everywhere with respect to Lebesgue measure.

Ans. Take to be the characteristic function of the measureable set Then implies that a.e. - the same argument works for so a.e.

Problem 2

Suppose is a compact operator on a Hilbert space and that has no positive eigenvalues, show that

Ans. Since is compact and selfadjoint there is a complete orthonormal basis of its eigenvectors. All the eigenvectors must be non-negative, since imples So, if there are no positive eigenvalues they must all be zero. Thus for all implies for all so

Problem 3

Give an example of a function which is continuous but is such that its Fourier transform

Ans. For any it is easy to find a non-negative continuous function, with maximum supported in The integral of its square is then less than Consider the function

All the terms have disjoint supports and is square integrable. If its Fourier transform was in the function would have to be continuous, which it is, and would also have to vanish infinity, which it does not. [This one is a bit tricky.]

Problem 4

Suppose and there exists such that

for all smooth -periodic functions, on the real line. Show that has a continuous representative in

Ans. Plug into the identity for each and you find that the Fourier coefficients of satisfy where are the Fourier coefficients of Thus

This is enough to imply that the Fourier series for converges uniformly so `is' continuous - has a representative which is continuous.

Problem 5

Suppose has Fourier transform satisfying in Show that there exists such that for all

Ans. The function in in is in and satisfies Thus if is the inverse Fourier transform of it is in and satisfies

Problem 6

Show that there is no element of satisfying

Ans. This stops the Fourier coefficietns of from vanishing at

Problem 7

Show that there exists a function such that

for all smooth -periodic functions on the real line.

Ans. The given condition implies he uniform convergence of the (formal) Fourier series for So has a continuous first derivative. Integration by parts is then justified in the given identity so we can take

Problem 8

If show that

 (45)

defines a compact operator

Ans. Change variables to shift the integral to (or use Fourier series on by setting The

Anyway, we get the same sort of integral with a different continuous function. Now, integrating agains we again find that the Fourier coefficients of satisfy whenever This implies that maps the ball into a compact set.

Problem 9

Show that there is an infinite orthonormal sequence with each element satifying

Ans. The eigenfunctions of the harmonic oscillator.

Richard B. Melrose 2004-05-24