Today’s problem appeared as Problem 3 on the UCLA Spring 2021 Analysis Qual:
Problem 3. For 
define
![\[f_n(x) := \int_0^{2\pi} f(x+t) \cos(nt)dt.\]](https://www.stepanmalkov.com/wp-content/ql-cache/quicklatex.com-0f87f697bdac2995652ce5123a45cdd4_l3.png "Rendered by QuickLaTeX.com")
a.e. and in 
Solution: Using the change of variables 
we note that 
where 
is the 
-th Fourier coefficient and 
is the 
-periodic function 
on ![[-x,2\pi-x].](https://www.stepanmalkov.com/wp-content/ql-cache/quicklatex.com-6b5eb1879a051de3215a774bea6d98da_l3.png "Rendered by QuickLaTeX.com")
Since the Fourier series of an 
function is in 
it follows that 
for all 
as 
which implies that 
pointwise everywhere.
b) We note that 
and evaluate the norm of 
using Fubini’s theorem:
![\[\int |f_n(x)|^2 dx = \int \int_{0}^{2\pi} f(x+t) e^{int} dt \overline{\int_0^{2\pi} f(x+t') e^{int'}dt'} dx\]](https://www.stepanmalkov.com/wp-content/ql-cache/quicklatex.com-ee18f6d908f4751f05ded89ca8d423ee_l3.png "Rendered by QuickLaTeX.com")
![\[= \int_0^{2\pi} \int_{0}^{2\pi} e^{in (t-t')} \int f(x+t)f(x+t') dx dt dt'.\]](https://www.stepanmalkov.com/wp-content/ql-cache/quicklatex.com-1a75dab5430172ec971645a6e5535b51_l3.png "Rendered by QuickLaTeX.com")
we note by Hölder’s inequality that ![g \in L^\infty([0,2\pi] \times [0,2\pi]).](https://www.stepanmalkov.com/wp-content/ql-cache/quicklatex.com-eba9c23d6cc0cfd96a450b3364699235_l3.png "Rendered by QuickLaTeX.com")
Performing the linear change of variables 
and noting that it has constant Jacobian, we thus rewrite ![\[\int |f_n(x)|^2 dx = C\int_0^{2\pi} \int_0^{2\pi} e^{ins} g(s,s') ds ds'.\]](https://www.stepanmalkov.com/wp-content/ql-cache/quicklatex.com-e56cbc885d1fd4abaf661181efefa032_l3.png "Rendered by QuickLaTeX.com")
for fixed 
and a.e. 
and 
for a.e. so by the Dominated Convergence theorem we have 
i.e. in