Fourier Series and Related Questions

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x) = \left|x\right|$ for $-\pi < x < \pi$.

Expand in the simplest Fourier series the function $f(x) = x$ for $-\pi < x < \pi$.

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x) = x^{2}$ for $-\pi < x < \pi$.

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x) = \left|\sin x\right|$ for $-\pi < x < \pi$.

Expand in the simplest Fourier series the function $f(x) = \cos ax$ for $-\pi < x < \pi$, $a$ not an integer.

Expand in the simplest Fourier series the function $f(x) = \sin ax$ for $-\pi < x < \pi$, $a$ not an integer.

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions: $f(x) = 0$ for $-\pi < x < 0$; $f(x) = \sin^{2} x$ for $0 < x < \pi$.

Expand in a Fourier series the periodic function: $f(x) = \left\{ x\right\}$, the fractional part of $x$, i.e. the difference $x-[x]$, where $[x]$ is the integer part of $x$.

Expand in a Fourier series the periodic function: $f(x) = (x)$, the distance from $x$ to the nearest integer.

Expand in a Fourier series the periodic function: $f(x) = x$, $1$, or $3-x$, according as $x$ lies in the interval $(0,1)$, $(1,2)$, or $(2,3)$, assuming that the period of $f(x)$ equals $3$.

If the function is not given as periodic and one seeks an expansion valid only on a given interval, then it can be expanded in a Fourier series with period equal to the length of the given interval, or with any larger period, but in the latter case the function must be extended in some way (as long as the Dirichlet conditions are satisfied). This can be used to obtain an expansion in cosines alone or in sines alone, or to restrict oneself to only the sines or cosines of odd multiples of the angle.

In the problems that follow, it will be assumed that the period is to be chosen as the smallest one for which the requirements of the problem are satisfied.

On the interval $(-h,h)$, expand the function $e^{x}$.

Expand $\operatorname {ch}ax$ on the interval $(-\pi ,\pi )$.

Expand $\operatorname {sh}ax$ on the interval $(-\pi ,\pi )$.

On the interval $(0,\pi )$, expand $x^{3}$ in cosines.

On the interval $(0,\pi )$, expand $x^{3}$ in sines.

On the interval $(0,\pi )$, expand $\sin x$ in cosines.

The function $f(x) = x^{2}(a-x)$ is given on the interval $(0,a)$. Expand it in a Fourier series with period $a$.

The function $f(x) = x^{2}(a-x)$ is given on the interval $(0,a)$. Expand it in a Fourier series as an even function.

The function $f(x) = x^{2}(a-x)$ is given on the interval $(0,a)$. Expand it in a Fourier series as an odd function.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series with period $a$.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series as an even function.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series as an odd function.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series as an odd function, in sines of odd multiples of the angle.

Functions rational in $\cos x$ and $\sin x$ can be expanded in a Fourier series without using the Fourier formulas. To do this we write:

where $t = e^{xi}$. The problem then reduces to expanding the function in positive and negative powers of $t$. Passing back from $t$ to $x$, we obtain the desired Fourier series expansion.

By the method just described and similar ones, obtain the following expansions:

By the method just described, obtain the following expansion:

By the method just described, obtain the following expansion:

By the method just described, obtain the following expansion:

By the method just described, obtain the following expansion:

The function $f(x)$, given on the interval $\left(0,\dfrac {\pi }{2}\right)$, is continued onto the interval $\left(\dfrac {\pi }{2},\pi \right)$, so that $f\left(\dfrac {\pi }{2}+x\right) = -f\left(\dfrac {\pi }{2}-x\right)$; prove that for $0 < x < \pi$ the equality holds

where

How should the function $f(x)$, given on the interval $\left(0,\dfrac {\pi }{2}\right)$, be continued onto the interval $\left(\dfrac {\pi }{2},\pi \right)$ so that for $0 < x < \pi$

and how can the coefficients be computed?

Prove, for $0 < x < \pi$, the following equalities:

Prove, for $0 < x < \pi$, the following equalities:

Prove the equality:

Prove the equality:

Prove the correctness of the expansion:

on the interval $0 < x < \dfrac {\pi }{2}$.

Prove the equalities:

Prove the equalities:

Prove the equalities:

Prove the equalities:

Prove the equality (Poisson's formula):

The modulus of continuity $\omega (h)$ is the exact upper bound of the oscillation of the function $f(x)$ on intervals of length $h$, i.e. of the numbers $\left|f(x_{2})-f(x_{1})\right|$, where $\left|x_{2}-x_{1}\right| < h$. Prove that for a continuous function $f(x)$ the integrals

are, in absolute value, no greater than $C\omega \left(\dfrac {\pi }{n}\right)$, where $C$ is a corresponding constant.

Prove the equality:

where $O(1)$, as $m \rightarrow \infty$, is a bounded quantity.

Prove Lebesgue's inequality:

Here $a_{n}$ and $b_{n}$ are the Fourier coefficients of the function $f(x)$, and $M$ is an upper bound for $\left|f(x)\right|$.

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

Prove that for any real $x$ and integer $n > 0$ the equality holds

For $0 < x < \pi$, prove the equality:

For any real $x$ and integer $n > 0$, prove the equality:

Prove Fejér's inequality:

The Fejér sum $\sigma_{n}$ is the arithmetic mean of the first $n$ Fourier sums of the function $f(x)$:

where for $m \geqslant 1$

$a_{\nu }$, $b_{\nu }$ being the Fourier coefficients of the function $f(x)$. Prove that $\left|\sigma_{n}(x)\right| < M$, if $\left|f(x)\right| < M$ throughout the period.

Prove that if $f(x)$ is a continuous function on the interval $\left[-\pi ,\pi \right]$, $l \leqslant f(x) \leqslant L$, and for every $m$, $\left|ma_{m}\right| < A$, $\left|mb_{m}\right| < B$, then

where $a_{m}$, $b_{m}$ are the Fourier coefficients, $s_{n}(x)$ is the $n$-th partial sum of the Fourier series of $f(x)$.

Prove that for every continuous function $f(x)$ on the interval $\left[-\pi ,\pi \right]$, $\sigma_{n}(x) \rightarrow f(x)$ as $n \rightarrow \infty$.

Prove that for a periodic function $f(x)$ satisfying the Cauchy--Lipschitz condition

the inequality $\left|\sigma_{n}-f(x)\right| < \dfrac {A\ln n}{n}$ holds, where $A$ is a constant.

Taking $f(x)$, wherever it occurs, to be a continuous function with period $2\pi$, prove the following formulas and assertions:

The integral $\dfrac {2}{\pi } \cdot \dfrac {(2n)!!}{(2n-1)!!} \int_{-\pi }^{\pi } f(x+u) \cos^{2n}\dfrac {u}{2}\, du$ is a trigonometric polynomial that tends to $f(x)$ as $n \rightarrow \infty$.

Prove that

where $F_{m}(u) = \left(\dfrac {\sin mu}{m\sin u}\right)^{4}$, and $c$ is some constant.

The quantity $T_{m}(x)$, where

($F_{m}(u)$ and $A_{m}$ are defined in problem 4915) is a trigonometric polynomial of order $2m-2$.

Prove the equality:

For the value of $A_{m}$, see problem 4915.

If $f(x)$ satisfies the condition

then Jackson's inequality holds:

where $M$ is a corresponding constant.

Let $f(x)$ be a function integrable on the interval $(0,2\pi )$, let $a_{n}$ and $b_{n}$ be its Fourier coefficients, let $S_{m} = \dfrac {a_{0}}{2}+\sum_{n=1}^{m} (a_{n}\cos nx+b_{n}\sin nx)$ be its Fourier sum, and let $P_{m}(x)$ be any trigonometric polynomial of order $m$, different from $S_{m}(x)$. Prove the inequality:

Prove Bessel's inequality:

Starting from Bessel's inequality and the inequality $2ab \leqslant a^{2}+b^{2}$, prove the convergence of the series $\sum_{\nu =1}^{\infty } (a_{\nu }^{2}+b_{\nu }^{2})$ and $\sum_{\nu =1}^{\infty } \dfrac {\left|a_{\nu }\right|+\left|b_{\nu }\right|}{\nu^{\alpha }}$, where $\alpha > \dfrac {1}{2}$.

Prove the inequality:

Let $f(x)$ be a function integrable on $(0,2\pi )$, let $\varepsilon$ be an arbitrary positive constant, let $\varphi (x)$ be a continuous function such that $\int_{0}^{2\pi } \left|f(x)-\varphi (x)\right|^{2}\, dx < \varepsilon$, and let $\sigma_{n}(x)$ be the Fejér sum (see problem 4910) for $\varphi (x)$, differing from $\varphi (x)$ by less than $\sqrt{\dfrac {\varepsilon }{2\pi }}$. Prove the inequality:

Using the preceding results, prove A.M. Lyapunov's theorem (the completeness theorem):

valid for any integrable function $f(x)$.

Prove the equality:

where $\varphi (x)$ and $\psi (x)$ are integrable on the interval $(0,2\pi )$, and $a_{\nu }$, $b_{\nu }$, $\alpha_{\nu }$, $\beta_{\nu }$ are the Fourier coefficients of these functions.

Prove V.A. Steklov's theorem:

Prove V.A. Steklov's inequalities:

Prove the equality:

where $\sigma_{n}(x)$ is the Fejér sum of the function $f(x)$ (see 4910).

Prove that for a function $f(x)$ satisfying the Cauchy--Lipschitz condition

the inequality holds:

where $k$ is a corresponding constant.

Prove S.N. Bernstein's theorem: the Fourier series of a function satisfying the Cauchy--Lipschitz condition with $\alpha > \dfrac {1}{2}$ converges absolutely.

The parametric equation of a closed curve can be put in the form

Using the formulas $ds^{2} = dx^{2}+dy^{2}$ and

where $A$ is the area bounded by the closed curve $S$, prove the inequality $4\pi A \leqslant L^{2}$ ($L$ being the length of the curve $S$), which expresses that among all closed curves of length $L$ the circle has the greatest area (Hurwitz's proof).

Expand in Chebyshev polynomials $T_{n}(x)$ the function defined on the interval $(-1,1)$ by the condition $f(x) = x^{3}$.

Expand in Chebyshev polynomials $T_{n}(x)$ the function defined on the interval $(-1,1)$ by the conditions:

Expand in Chebyshev polynomials $T_{n}(x)$ the function defined on the interval $(-1,1)$ by the condition $f(x) = \left|x\right|$.

Expand in Chebyshev polynomials $T_{n}(x)$ the function defined on the interval $(-1,1)$ by the condition $f(x) = x^{2n}$.

Expand, for $-1 < x < 1$, in Legendre polynomials the function defined by the conditions:

Expand, for $-1 < x < 1$, in Legendre polynomials the function defined by the condition $f(x) = \left|x\right|$.

Expand in Laguerre polynomials $L_{n}(x)$, for $x > 0$, the function $f(x) = e^{-ax}$.

Expand in Laguerre polynomials $L_{n}(x)$, for $x > 0$, the function $f(x) = x^{n}$.