This test is closed book. You are not permitted to bring any books, notes or such material with you. You may use theorems, lemmas and propositions from the book or from class.

Note that most of the solutions are relatively short - this is likely to be the case in the final as well!

- If
is a convergent sequence, with respect to the
norm show that there is a subsequence which converges
pointwise almost everywhere.
Solution:- Since has finite measure we know that and in fact

- Suppose that
has Fourier coefficients
satisfying

Show that is continuous onSolution:- Since we know that its Fourier series converges to in

- Let
be the sequence obtained by
orthonormalization (the Gramm-Schmid process) of
Show that for each the Fourier transform
is linearly
dependent on
Solution:- The Gram-Schmid process replaces the by where each is a linear combination of the for Thus it is enough to show that itself is linearly dependent on the Fourier transforms Now, we also know that for some non-zero constant so the statement is true for We can proceed by induction, assuming that we have already shown that the statement is true for and then just prove it for In fact the Fourier transform satisfies

- Show that if
is a bounded
measurable function which satisfies
for
all non-negative integers
then there is an odd function
such that for almost all
Solved:- Since is bounded and measurable, it is in Consider the function This is even and in Moreover it satisfies

Now, we know that polynomials are dense in so we can choose a sequence in Thus it follows that in and hence thatalmost everywhere.That almost everywhere implies that

Richard B. Melrose 2004-05-24