Fourier coefficients under symmetry and operations

Prove that $\hat{\overline{f}}(n) = \overline{\hat{f}(-n)}$ for all $f \in L^{1}(\partial \mathbf{D})$ and all $n \in \mathbf{Z}$.

Suppose $f \in L^{1}(\partial \mathbf{D})$. Prove that $f$ is real valued (almost everywhere) if and only if $\widehat{f}(-n) = \widehat{f}(n)$ for every $n \in \mathbf{Z}$.

Let $f(x)$ be an anti-periodic function with period $\pi$, that is,

What special property does the Fourier series of this function on the interval $(-\pi ,\pi )$ have?

Let $f(x+\pi )=f(x)$. What special property does the Fourier series of the function $f(x)$ on the interval $(-\pi ,\pi )$ have?

A function $y=f(x)$ with period $2\pi$ has a graph that is:

If the functions satisfy $\varphi (-x)=\psi (x)$, what relation holds between the Fourier coefficients $a_{n},b_{n}$ of $\varphi (x)$ and $\alpha_{n},\beta_{n}$ of $\psi (x)$ ($n=0,1,2,\cdots$)?

If the functions satisfy $\varphi (-x)=-\psi (x)$, what relation holds between the Fourier coefficients $a_{n},b_{n}$ of $\varphi (x)$ and $\alpha_{n},\beta_{n}$ of $\psi (x)$ ($n=0,1,2,\cdots$)?

Given that the integrable function $f(x)$ with period $2\pi$ has Fourier coefficients $a_{n},b_{n}$ ($n=0,1,2,\cdots$), compute the Fourier coefficients $\overline{a_{n}},\overline{b_{n}}$ ($n=0,1,2,\cdots$) of the "shifted" function $f(x+h)$ ($h=$ constant).

Given that the integrable function $f(x)$ with period $2\pi$ has Fourier coefficients $a_{n},b_{n}$ ($n=0,1,2,\cdots$), compute the Fourier coefficients $A_{n},B_{n}$ ($n=0,1,2,\cdots$) of the Steklov function

Let $f(x)$ be a continuous function with period $2\pi$, and let $a_{n},b_{n}$ ($n=0,1,2,\cdots$) be its Fourier coefficients. Find the Fourier coefficients $A_{n},B_{n}$ ($n=0,1,2,\cdots$) of the convolution function

Using the result obtained, derive Lyapunov's equality.

Prove Lemma 13.5, and also derive analogous formulas for the inverse Fourier transforms $\left(T_{a} f\right)^{\vee },\left(M_{m} f\right)^{\vee },\left(T_{m} c\right)^{\vee }$, and $\left(M_{a} c\right)^{\vee }$.

Given $f \in L^{1}(\mathbf{T})$, set $f^{*}(x) = f(-x)$ and show that $\widehat{f^{*}}(n) = \widehat{f}(-n) = \stackrel{\vee }{f}(n)$. Conclude that if $f$ is even then $\widehat{f}$ is even and $\widehat{f} = \stackrel{\vee }{f}$.

What can be said about the Fourier coefficients of a function $f$ on $\mathrm{T}$ if it is known that $f$

Express the following properties of a function $f$ in terms of the Fourier coefficients:

Let the function $f$ be integrable on $[0,1/4]$. Extend $f$ to $[0,1]$ in such a way that its Fourier coefficients satisfy the relations $c_{2k}=0$, $c_{2k-1}=-c_{1-2k}$.

Let $\left\{ c_n\right\}$ be the Fourier coefficients of a function $f\in L^{1}\left({\mathrm{T},dt}\right)$. Find the Fourier coefficients $\left\{ c_n(h)\right\}$ for the smoothed function (or Steklov function) $f_h(x)=(1/2h)\int_{x-h}^{x+h}f(\xi )\, d\xi$.

Prove that if a function $f$, absolutely integrable on the segment $[-\pi ; \pi ]$, satisfies the condition:

What features do the Fourier coefficients of a function of period $2 \pi$ have if its graph:

How are the Fourier coefficients $a_{n}, b_{n}$ and $\alpha_{n}$, $\beta_{n}$ of the functions $f$ and $g$ related, if $f(-x) = g(x)$?

How are the Fourier coefficients $a_{n}, b_{n}$ and $\alpha_{n}$, $\beta_{n}$ of the functions $f$ and $g$ related, if $f(-x) = -g(x)$?

Prove that the Fourier series of a finite linear combination of $2 \pi$-periodic functions, absolutely integrable over the period, equals the same linear combination of the Fourier series of the given functions.

Let $f$ be a $2 \pi$-periodic function, absolutely integrable over the period. Prove that if $a_{n}^{*}, b_{n}^{*}$ are the Fourier coefficients of the function $f^{*}(x) = f(x+h)$, and $a_{n}, b_{n}$ are the Fourier coefficients of the function $f$, then

Prove that $f(x+\alpha ) \sim \sum_{n = -\infty }^{\infty }\left(c_{n} e^{i n \alpha }\right) e^{i n x}, \quad \alpha \in R$.

Prove that if $k \in Z$, then $f(x) e^{i k x} \sim \sum_{n = -\infty }^{\infty } c_{n-k} e^{i n x}$.

Prove that $\bar{f} \sim \sum_{n = -\infty }^{\infty } \bar{c}_{-n} e^{i n x}$.

Let $f(\vartheta )$ ($\vartheta$ real) be a real-valued periodic function, $f(\vartheta + 2\pi ) = f(\vartheta )$. Each of the following equations (or pairs of equations respectively) characterizes a special symmetry property of the graph of the curve $y = f(x)$: