Trigonometric Fourier Series

Expand the function into a Fourier series:

What will the Fourier series be for the trigonometric polynomial

Find the partial sum $S_{3}(x ; f)$ of the Fourier series of the function

periodically extended with period $2 l$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = x,-\pi \leqslant x \leqslant \pi , x_{0} = \pi$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \begin{cases} 1, & 0 \leqslant x \leqslant \pi , \\ 0, & -\pi \leqslant x < 0 ; \end{cases} \quad x_{0} = 0$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \begin{cases} \pi / 4, & 0 \leqslant x \leqslant \pi , \\ -\pi / 4, & -\pi \leqslant x < 0 ; \end{cases} \quad x_{0} = 0$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \begin{cases} a, & 0 \leqslant x \leqslant \pi , \\ -a, & -\pi \leqslant x < 0 ; \end{cases} \quad x_{0} = 0$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \pi +x,-\pi \leqslant x \leqslant \pi , x_{0} = \pi$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \left|x\right|,-\pi \leqslant x \leqslant \pi , x_{0} = \pi$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \begin{cases} 0, & -\pi \leqslant x < 0, \\ x, & 0 \leqslant x \leqslant \pi ; \end{cases} \quad x_{0} = -\pi$.

Expand the function $f(x)$ into a Fourier series, indicate the intervals on which the sum of the Fourier series equals the function $f(x)$, and find the sum of the series at the given point $x_{0}$. $f(x) = \begin{cases} -2 x, & -\pi < x \leqslant 0, \\ 3 x, & 0 < x \leqslant \pi ; \end{cases} \quad x_{0} = \pi$.

Expand the function $f(x) = \operatorname {sign} x,-\pi < x < \pi$, into a Fourier series and, using the resulting expansion, find the sum of the Leibniz series

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} A, & 0 < x < l, \\ A / 2, & x = l, \\ 0, & l < x < 2 l, \end{cases} \quad$ on the interval $(0,2 l)$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \left|x\right|$ on the segment $[-1 ; 1]$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} a x, & -\pi < x < 0, \\ b x, & 0 \leqslant x < \pi , \end{cases} \quad$ on the interval $(-\pi , \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} a, & -\pi / 2 < x < \pi / 2, \\ b, & \pi / 2 \leqslant x < 3 \pi / 2, \end{cases}$ on the interval $(-\pi / 2,3 \pi / 2)$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x+\operatorname {sign} x$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \pi^{2}-x^{2}$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x^{3}$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = e^{a x}, a \neq 0$, on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = e^{2\left|x\right|}$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \sin a x, a \in Z$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \cos a x, a \notin Z$ on the segment $[-\pi ; \pi ]$. Using the resulting expansion, prove that

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x \sin x$ on the segment $[-\pi ; \pi ]$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x \cos x$ on the segment $[-\pi / 2 ; \pi / 2]$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} 0, & -\pi \leqslant x < 0, \\ \sin x, & 0 \leqslant x \leqslant \pi , \end{cases}$ on the segment $[-\pi , \pi ]$.

Expand into a Fourier series the periodic function $f$ with period 2, if $f(x) = x, x \in (1 ; 3)$. Draw the graph of the sum of the series.

Expand into a Fourier series the function $f(x) = x^{2},-1 \leqslant x \leqslant 1$, periodically extended with period 2. Draw the graph of the sum of the series.

Expand the function into a Fourier series

periodically extended to the entire real line with period 3. Draw the graph of the sum of the series.

Expand into a Fourier series the function $f(x) = \pi -2 x, 0 < x \leqslant \pi$, extending it to the segment $[-\pi ; 0]$:

Expand into a Fourier series the function $f(x)$, given on the segment $[0 ; \pi ]$ and extended to the segment $[-\pi ; 0]$ evenly. Draw the graph of the sum of the series:

Expand the periodic function into a Fourier series. $f(x) = \operatorname {sign}(\cos x)$.

Expand the periodic function into a Fourier series. $f(x) = \arcsin (\cos x)$.

Expand the periodic function into a Fourier series. $f(x) = \arcsin (\sin x)$.

Expand the periodic function into a Fourier series. $f(x) = x-[x]$.

Expand the periodic function into a Fourier series. $f(x) = (x)$ is the distance from $x$ to the nearest integer.

Expand the periodic function into a Fourier series. $f(x) = \left|\cos x\right|$.

Expand the periodic function into a Fourier series. $f(x) = \left|\sin x\right|$.

Expand the periodic function into a Fourier series. $f(x) = \left|\cos (x / 2)\right|$.

Expand the function $f(x) = x, 0 \leqslant x \leqslant \pi$, into a Fourier series in cosines.

Expand the function $f(x) = \cos 2 x, 0 \leqslant x \leqslant \pi$, into a Fourier series in sines.

Expand into a Fourier series on $(0 ; \pi )$ in cosines the function

Expand into a Fourier series on the interval $(0 ; \pi )$ in sines the function

Expand the function

into a Fourier series in cosines on the segment $[0 ; 2]$.

Expand the function $f(x) = x^{2}$ into a Fourier series:

Expand into a Fourier series the function $f(x) = x-x^{2} / 2,0 \leqslant x \leqslant 1$:

Expand into a Fourier series in sines the function

Prove that $\frac{3 x^{2}-6 \pi x+2 \pi^{2}}{12} = \sum_{n = 1}^{\infty } \frac{\cos n x}{n^{2}}, 0 \leqslant x \leqslant \pi$.

Hint. Use the results of problems 40 and 45.

Expand into a Fourier series in cosines the function $f(x) = e^{x}$ on the interval $(0 ; \ln 2)$.

Expand into a Fourier series in sines the function

Expand into a Fourier series in sines the function

Expand the function $f(x) = e^{a x}, 0 < x < \pi$:

Expand the function $f(x) = \sin a x, 0 \leqslant x \leqslant \pi$, into a Fourier series in cosines.

Expand into a Fourier series in cosines the function

Expand into a Fourier series in cosines the function

Prove that

Find the Fourier series in complex form of the function

Expand into a Fourier series in complex form the periodic function, with period $2 \pi$, $f(x) = e^{x},-\pi < x < \pi , f(\pi ) = \operatorname {ch} \pi$.

Expand into a Fourier series in complex form the periodic function with period 3

Expand the function into a Fourier series using the complex form of the Fourier series:

Expand into a Fourier series the unbounded periodic function:

Using the expansion of the function $f(x) = x$ into a Fourier series (see problem 4), prove that:

Using the expansion of the function $\ln \left|-2 \cos (x / 2)\right|$ into a Fourier series (see example 6), prove that:

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

Prove that if a function $f$, absolutely integrable on the segment $[0 ; \pi ]$, satisfies the condition $f(\pi -x) = f(x)$, then its Fourier coefficients have the following properties:

How should a function, absolutely integrable on the segment $[0 ; \pi / 2]$, be extended to the segment $[-\pi ; \pi ]$ so that its Fourier series has the form

How should a function, absolutely integrable on the segment $[0 ; \pi / 2]$, be extended to the segment $[-\pi ; \pi ]$ so that its Fourier series has the form

Expand the function $f(x) = x(\pi / 2-x)$ into a Fourier series on the segment $[0 ; \pi / 2]$:

Prove that a piecewise smooth function on the segment $[0 ; l]$ can be expanded into a series of the form

Prove that a piecewise smooth function on the segment $[0 ; l]$ can be expanded into a series of the form

Prove the equality for $0 < x < \pi$:

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 the functions $f$ and $g$ be $2 \pi$-periodic, with $|f|^{2}$ and $|g|^{2}$ integrable (in the improper sense) on the segment $[-\pi ; \pi ]$,

Prove that then the product $f g$ is a $2 \pi$-periodic function, absolutely integrable over the period, and if

Prove that

Prove that for any $x \in R, x \neq \pi m, m \in Z$, the equality holds

Prove that if the function $f$ is continuously differentiable on the segment $[x ; x+\pi ]$, then

(Steklov).

Let $a_{n}, b_{n}$ be the Fourier coefficients of the function equal to $e^{x}$ on the interval $(-\pi ; \pi )$. Find the sum of the series

for $x \in (-\pi ; \pi )$.

Prove that the Fourier coefficients $a_{n}, b_{n}$ of a function absolutely integrable on the segment $[-\pi ; \pi ]$ tend to zero as $n \rightarrow \infty$ (Riemann).

Prove that the Dirichlet kernel $D_{n}$ is an even, continuous, $2 \pi$-periodic function, $D_{n}(0) = n+1 / 2$,

and, for $t \neq 2 \pi k, k \in Z$, the equality holds

Prove that if $f$ is a $2 \pi$-periodic function, absolutely integrable over the period, and $S_{n}(x)$ is its Fourier sum of order $n$, then:

Prove that if $f$ is a $2 \pi$-periodic function, absolutely integrable over the period, then the existence and value of the limit of the sequence of its Fourier sums $S_{n}(x)$ at every point $x_{0} \in R$ depends only on the existence and value of the limit, as $n \rightarrow \infty$, of the integral

where $\delta$ is an arbitrarily small positive number (this statement is called the localization principle for Fourier series).

Let $f$ be a $2 \pi$-periodic function, absolutely integrable over the period. Prove that if $x_{0}$ is a point of continuity or a discontinuity of the first kind and, for some $\delta > 0$, the integral converges

where

then the Fourier series of the function $f$ at the point $x_{0}$ converges to the value

(Dini's test).

Prove that if $f$ is a $2 \pi$-periodic function, absolutely integrable over the period, and at the point $x \in R$ the limiting values $f(x+0), f(x-0)$ and the one-sided derivatives exist $f_{+}^{\prime }(x), f_{-}^{\prime }(x)$, then the Fourier series of the function $f$ converges at this point to the value $\frac{f(x+0)+f(x-0)}{2}$.

A function $f$, defined in some neighborhood $U$ of the point $x$, is said to satisfy the Hölder condition of order $\alpha > 0$ at this point if the one-sided limits $f(x+0), f(x- -0)$ exist, and there is a constant $c > 0$ such that for all $h > 0$ for which $x+h \in U$ and $x-h \in U$, the inequalities hold

Prove that if a $2 \pi$-periodic function $f$, absolutely integrable over the period, satisfies the Hölder condition of order $\alpha > 0$ at the point $x$, then its Fourier series converges at this point and its sum equals

Prove that if a $2 \pi$-periodic function, absolutely integrable over the period, has a bounded derivative in a neighborhood of a given point, then its Fourier series converges at this point to the value of the function.

Prove that the Fejér kernel $\Phi_{n}$ is an even, continuous, $2 \pi$-periodic function,

and, for $t \neq 2 \pi k, k \in Z$, the equality holds

Prove that the Fejér kernel $\Phi_{n}$ satisfies, for any $\delta > 0$, the condition

Prove: if a function is continuous on the segment $[-\pi ; \pi ]$ and takes equal values at its endpoints, then the sequence of its Fejér sums converges uniformly on this segment to the function in question.

Prove that if $f$ is a $2 \pi$-periodic function, absolutely integrable over the period, having left- and right-hand limits $f(x \pm 0)$ at the point $x$, then (see notation (18))

Prove that if a $2 \pi$-periodic function $f$, absolutely integrable over the period, is continuous at the point $x$ and the Fourier series of $f$ converges at this point, then its sum equals $f(x)$.