Before getting into the question, just to let you guys know that I have a final tomorrow and this was on the past test but I'm lost as to how to proceed on this, so any help will be great!
Edit - I see people have said that this question needs additional content, i'm not sure what I can provide. As mentioned earlier, This is from a past final exam paper for a Real Analysis class, and was referred as good practice by the professor, though nothing else was said about it. I get that this question is a little broad in terms of the topic, but I'm looking for outright answers to the question but just the thinking behind the answer like Doug has mentioned below. I wouldn't be asking this if I had another option.
Consider a $C^0_{2π-periodic}$ of continuous functions over $R$ with period $2π$. Let us consider a map T which acts as a functions on $C^0_{2π-periodic}$ and produces new functions (T(f))(x) defined via the formula:
(T(f))(x) = $\int_x^{x+π/6} f(t)dt$
(a) Show that for every f $\epsilon$ $C^0_{2π-periodic}$, T(f) also lies in in the space $C^0_{2π-periodic}$.
(b)Thinking of the space $C^0_{2π-periodic}$ as a metric space (with the usual $C^0$ norm) show that the map T is continuous with respect to the $C^0$ metric on this space.
(c) Show that if f(x) is a trigonometric polynomial of order m, $_f(x)$ = $\sum_{k=−m}^m$ $a_me^{im·x}$ then T(f) is also a trigonometric polynomial of order m. Find an explicit expression for T(f) as a trigonometric polynomial in this case.
(d) Derive that there exists a constant C > 0 so that for every n $\epsilon$ Z, n ≠ 0 the $n^{th}$ Fourier coefficient $\widehat{T(f)}$ of T(f) satisfies |$\widehat{T(f)}$ (n)| ≤ _C|$\hat{f}$(n)| · $|n|^{−1}$ and |$\widehat{T(f)}$(0)| $\le$ C|$\hat{f}$(0)
(e) Consider the space B of 2$\pi$-periodic and Riemann-integrable functions R($\Bbb{R}$) for which $\int_{-\pi}^{\pi}|f(x)|^2 dx \le 1$
Consider the space of sequences
{${....\widehat{T(f)}(-n-1),$\widehat{T(f)}(n)},...,\widehat{T(f)}(0),...,\widehat{T(f)}(n),\widehat{T(f)}(n+1),...$}
where $f \varepsilon$ B. Show the closure of this space in $\ell^2 (\Bbb{Z})$ is compact.
To start with, if it is continuous and periodic, it can be modeled as a Fourier series. This is suggested in the later parts of the question.
a) is $T(f(x))$ periodic?
$T(f(x+2\pi) = \int_{x+2\pi}^{x+\frac {13}{6}\pi} f(t)\ dt\\ u = t-2\pi\\ \int_{x}^{x+\frac {1}{6}\pi} f(u-2\pi)\ du\\ \int_{x}^{x+\frac {1}{6}\pi} f(u)\ du = T(f(x))$
Is it continuous?
$|T(f(x+\delta)) - T(f(x))| = |\int_{x}^{x+\delta} f(t)\ dt|+ |\int_{x+\frac 16 \pi}^{x+\frac 16 \pi + \delta} f(t)\ dt|$
And can you show that these are less than delta.
Or you could look at the Fourier series.
$\int_x^{x+\frac 16\pi} e^{imx}$ continous and differentiable for all natural $m$
Sorry, I don't have time for a more complete answer, but does that get you started?