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Questions tagged [periodic-functions]

Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.

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Linking Fourier Coefficients of periodic functions

Let $\tau\in (0,1)$ and assume that we have a $\tau$-periodic function $$f_1(t) = \sum\limits_{k\in\mathbb{Z}} a^1_k e^{\frac{2\pi i k}{\tau}t},$$ a $(1-\tau)$-periodic function $$f_2(t) = \sum\...
Nuke_Gunray's user avatar
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4 votes
2 answers
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Periodic perturbation of ODE

Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.5, the author mentioned that the following is well-known in the theory of ODEs. If $x_0$ is an ...
R-CH2OH's user avatar
  • 195
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0 answers
35 views

Prove that, If a function is symmetric about two different lines perpendicular to the axis of $x$, then it is periodic.

Consider a real continous function $f$ for all $x \in \mathbb{R}$ be symmetric about two lines perpendicular to the axis of $x$ (say $x=a,x=b, a>b$) $\therefore$ \begin{equation} \tag{1} f(a-x)=f(a+...
Jesko's user avatar
  • 45
0 votes
2 answers
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Period of function $f(x) = \{x\} + \{x + \frac{1}{3}\} + \{x + \frac{2}{3}\}$

Consider the function $f(x) = \{x\} + \{x + \frac{1}{3}\} + \{x + \frac{2}{3}\}$ Let's break this into three individual functions $g(x) = \{x\}$, $h(x) = \{x + \frac{1}{3}\}$, and $i(x) = \{x + \frac{...
Haider's user avatar
  • 59
1 vote
0 answers
37 views

Uniform convergence and continuous limit in $C^0$ and $L^{\infty}$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be $2\pi$-periodic and $f_n \in C^0_{2\pi}(\mathbb{R})$ for all $n \in \mathbb{N}$ such that $\lim_{n \rightarrow \infty} \| f_n-f\|_{\infty} =0$. Then $f \...
Oscar210899's user avatar
2 votes
1 answer
104 views

Analytical transformation or an effective numerical method for calculating $\sum_{n=1}^{\infty}{K_0(\frac{n}{a})\sin(n)}$ series

I have a series for that I need to get a fast calculation: $$\sum_{n=1}^{\infty}K_0\!\left(\frac{n}{a}\right)\sin(n)$$ where $K_0$ is the $0^{\text{th}}$ modified Bessel functions of the second kind, ...
gearquicker's user avatar
1 vote
0 answers
29 views

An equality for a $1$ periodic continuous function.

Let $f: \Bbb{R} \to \Bbb{R}$ be continuous such that $f(x+1)=f(x)$ for all $x \in \Bbb{R}$. Fix some $a \in \Bbb{R} \setminus \Bbb{Q}$ so an irrational number. Prove $$\lim_{n \to \infty} \frac{1}{n} \...
homosapien's user avatar
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Question regarding period of functions resulting from algebraic operations of functions

The method to find the period of functions resulting from algebraic operations of functions in my textbook is as follows Say f(x) has period $p = m/n$($m, n \in N$ and coprime) and f(x) has period $q =...
koiboi's user avatar
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1 vote
1 answer
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Can we show that the following integral is independent of $\phi_0$ without computing the integral itself?

$$\int_0^{2\pi N} d\phi \sin^4\left(\dfrac{\phi}{2N}\right)\sin^2\left(\phi+\phi_0\right)$$ Here $N$ is an integer greater than $1$ and $\phi_0$ is a constant phase. Context: while studying laser ...
Reshad's user avatar
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4 votes
0 answers
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How do we measure the "holes" (zeros) of the set of $\Bbb{Z}$-linear combinations of $f_i : G \to G$ where $G$ is an abelian group?

Question: Let $G$ be a normed abelian group. Namely the triangle inequality holds $|g + h|\leq |g| + |h|$. An example would be $G = \Bbb{Z}$ together with $|\cdot| =$ absolute value. Now suppose ...
HighAsAKiteOnMath's user avatar
6 votes
2 answers
232 views

Is it true that $\sin^2((2n+1) \pi x) \geq c$ occurs often (in precise sense)?

Fix $x \in [0, 1/2]$ and consider the function $f(n, x) = \sin^2((2n +1) \pi x)$. Based on plotting, it seems like the following is true For every $x \in (0, 1/2)$, there exists $c \in (0, 1]$ and a ...
Drew Brady's user avatar
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3 votes
2 answers
48 views

General form of functions with $\hat{f}(n) = 0$ for all $|n| \geq 2$

How to determine the general form of functions with $\hat{f}(n) = 0$ for all $|n| \geq 2$? Here $\hat{f}(n)$ denotes the $n$-th Fourier coefficient of Riemann integrable function $f$. It is also ...
schneiderlog's user avatar
2 votes
0 answers
58 views

Functional equation involving periodic functions and derivative [closed]

Let $n \in \mathbb{N}_0$, and $w > 0$ be a real number. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an even differentiable function, and periodic of period $w$. Show that if $f$ satisfies $$f(x) +...
Dave's user avatar
  • 33
3 votes
0 answers
102 views

Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$

problem statement: Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$ but the Fourier series of $f^2(x)$ is uniformly ...
Martin.s's user avatar
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1 vote
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Is it possible to find a solution to ODEs assuming the solution is periodic with known period?

I have a nonlinear system of ODEs with known constant coefficients $A, B, C, D, E, F, M$: \begin{align} &\dot{n}(t)=-An(t)+Bm(t)n(t)+Cm(t) \\ &\dot{m}(t)=-Bm(t)n(t) + (M-m(t))R_0 \sin{\omega t}...
Andris Erglis's user avatar
1 vote
1 answer
66 views

Let $F:\Bbb{R} \to \Bbb{R}$ be a positive, smooth and periodic function with period $P>0$. Proof that...

Let $F:\Bbb{R} \to \Bbb{R}$ be a positive, smooth and periodic function with period $p>0$. Prove that if $\Phi(t)$ is a solution to the differential equation $x'=F(x)$ and $$ T=\int_0^p {1\over F(y)...
Mauricio Rosendo Montiel's user avatar
1 vote
1 answer
69 views

The uniqueness of the Airy equation, which is a third-order linear partial differential equation.

For the Airy equation, which is a third-order linear partial differential equation given by: $$ \partial_{t}u+\partial_{xxx}u=0.$$ Suppose furthermore that we consider solutios of Airy equation which ...
Lilili123's user avatar
  • 139
2 votes
2 answers
143 views

An exotic operation on repeating sequences of natural numbers such as $\overline{6,9} = 6,9,6,9,6,9, \dots$ Having to do with common subsums.

Definitions. This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully ...
HighAsAKiteOnMath's user avatar
2 votes
1 answer
45 views

Showing the uniqueness of a solution to an ordinary differential equation [closed]

$$ \dot{y}(x)+y(x)=u(x) $$ $$ y_c(x) = ce^{-x} + \int_{0}^{x} u(t) e^{(t-x)} dt $$ Show that if u is periodic with a period of T, then there exists exactly one solution to the differential equation ...
Julian P's user avatar
0 votes
2 answers
55 views

Why the integral in $[-\pi-\frac{\pi}{k}, \pi-\frac{\pi}{k}]$ equals the integral in $[-\pi, \pi]$?

I'm showing that the coefficients of the Fourier series of a $2\pi$-periodic function $f$ can be written as $$a_k = \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \left(f(x) - f\left(x - \frac{\pi}{k}\right)\...
Cyclotomic Manolo's user avatar
0 votes
0 answers
35 views

On the structure of an integral with a variable upper limit of a continuous periodic function

$f$ is continuous and periodic with period T. Show that $F: x \to \int_0^xf(t)dt$ can be represented as a sum of a linear and a periodic function. I have proven that $\int_x^{x+T}f(x)dx=\int_0^Tf(x)dx$...
mathemaniac's user avatar
0 votes
1 answer
48 views

Does a periodic function on $\mathbb{R}^n$ have fixed point?

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. We assume that $f$ is periodic: for every $x \in \mathbb{R}^n$ and every $a \in \mathbb{Z}^n$ we find $f(x) = f(x + a)$. Is it true ...
Daniel Goc's user avatar
0 votes
1 answer
78 views

Prove/Disprove for $f:\mathbb N\to\mathbb N,$ if $f\circ f$ is periodic then $f$ is periodic

We define periodic according to the standard definition where $f(x)=f(x+p)$, this means that there exists p so that $f(f(x)) = f(f(x)+p)$. However I am unable to find a counter example or a ...
Anan Saadi's user avatar
1 vote
1 answer
42 views

Tilings closed under translation in only one direction

Background A tiling or tessellation of the plane is periodic if it is closed under at least two non parallel translations. Three examples of periodic tilings, including their corresponding translation ...
Max Muller's user avatar
  • 7,098
4 votes
1 answer
77 views

Periodic solutions to a first order linear ODE with incommensurable periods

I'm studying for an admission test for the PhD in Mathematical Analysis. I'm stuck to solve this exercise: Consider the ODE $x'(t)+a(t)x(t)=b(t)$ where $a(t)$ and $b(t)$ are continuous functions from $...
irbag's user avatar
  • 320
0 votes
0 answers
15 views

Nontrivial rational solutions for an anticommutative functional equation

I'm trying to find possible solutions for the following equation: $${ f\!\left(\matrix{x_1&x_2\\y_1+z_1&y_2+y_1z_1+z_2}\right) = f\!\left(\matrix{x_1&x_2\\y_1&y_2}\right) + f\!\left(\...
Aberone's user avatar
  • 212
1 vote
0 answers
32 views

Show the partition function $Q(n,k)$, the number of partitions of $n$ into $k$ distinct parts, is periodic mod m

Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts. I want to show that for any $m \geq 1$ there exists $t \geq 1$ such that $$Q(n+t,i)=Q(n,i) \mod m \quad \forall n>0 \forall ...
Kinkin's user avatar
  • 103
-1 votes
2 answers
50 views

My question is whether given function is periodic or not [closed]

F(x) = 1, if x is rational 0, if x is irrational So what i am really confused about is because all professors are telling me its periodic whose period is not defined, i mean it doesn't make any ...
MultiUniverseExplorer's user avatar
0 votes
1 answer
57 views

Existence of smallest period for continuous, periodic functions

I have found several partial answers to that question, all of which used more sophisticated mathematics than what I think is warranted by the question. Now, this may well be because my proofs below ...
wmnorth's user avatar
  • 597
0 votes
0 answers
40 views

Fourier transform on a regular lattice: The restricted set of wave-vectors

I) Consider a function defined only on the vertices of a 1D regular lattice: $f_i \equiv f(x_i)$ for all $x_i$, $i \in \{ 1, 2, ..., N \}$ and $x_{i+1} - x_i = a$, where $a > 0$ is the “lattice ...
AlQuemist's user avatar
  • 198
0 votes
1 answer
53 views

Weakly lower-semicontinuous functional on $L^2$

I am interested in the following problem: Is there any function (let's say continuous) $W:\mathbb{R}\to \mathbb{R}$ satisfying $W(x+1)=W(x) \forall x\in \mathbb{R}$ and such that the functional $L^2([...
Pong's user avatar
  • 23
2 votes
0 answers
73 views

Finding periodic solutions of a second degree differential equations

Let $w\in \mathbb R$ and $p>0$. I want to find the p-periodic solutions of the differential equation $$ f''(x)+w^2f(x)=0. $$ Let the $p$-periodic solution be $$ f(x)=c_0+\sum_{k \in \mathbb Z \...
palio's user avatar
  • 11.1k
1 vote
0 answers
34 views

Calculating Fourier coefficient of a split function with time period $y\left(x\right)=\begin{cases}-1&-T<x\le 0\\ 1&0<x\le T\end{cases}$

$$y\left(x\right)=\begin{cases}-1&-T<x\le 0\\ 1&0<x\le T\end{cases}$$ I tried to use definition: $B_k=\frac{1}{2T}\int _{-T}^{+T}y\left(x\right)e^{-jk\omega _0x}dx\:\:=\frac{1}{2T}\int _{...
Ben Shaines's user avatar
1 vote
0 answers
29 views

Floquet theorem for Hilbert spaces

Can the Floquet theorem be generalized to Hilbert spaces? I think the generalization would look something like this: Consider a dynamical system $\dot{x}(t)=A(t)x(t)$, where $A(t)$ is a family of ...
Riemann's user avatar
  • 717
0 votes
0 answers
78 views

How to determine the periodicity of this function?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfy $$f(x+y)f(x-y)=(f(x))^2+(f(y))^2-1,\forall x,y\in\mathbb{R}.$$ $\textbf{My question is: Is $f$ a periodic function}?$ In fact, let $y=0$, then $(f(0))^2=...
lkj123's user avatar
  • 46
0 votes
0 answers
57 views

Flaw in finding $f(x)$ when $f(0)=1$ and $f(10-x)=f(x), f(2-x)=f(2+x), \forall x \in R$

Finding, the function $f(x)$ when $$f(0)=1 ~\text{and}~ f(10-x)=f(x), f(2-x)=f(2+x), \forall x \in R.......(1)$$ Let $x \to 2-x$ in $f(10-x)=f(x) \implies f(8+x)=f(2-x)=f(2+x)$ Next, we take $x\to x-2$...
Z Ahmed's user avatar
  • 43.3k
4 votes
1 answer
231 views

Uniqueness of the writing $f(x)=cx+p(x)$, $p$ periodic

Show that, if a function $f:\mathbb{R}\rightarrow \mathbb{R}$, defined as $f(x)=cx+p(x)$, is the sum of a "linear part" $x\mapsto cx$ and a "periodic part" $x\mapsto p(x)$, where $...
100nanoFarad's user avatar
0 votes
0 answers
89 views

Number of subsequences of consecutive sequences in a non-periodic sequence

Let $a_1, a_2, \ldots$ be a sequence which is not eventually periodic, i.e. there do not exist constants $K$ and $N$ such that $a_m = a_{m+K}$ for all $m \geq N$. Prove that the number of distinct ...
DesmondMiles's user avatar
  • 2,713
2 votes
3 answers
211 views

Theorems about periodic functions

I'm trying to prove something about periodic functions and I'd need someone to tell me if what I wrote is right! If $f$ is a periodic function with fundamental period $\tau$. Then, all periods of $f$ ...
m05's user avatar
  • 23
1 vote
1 answer
68 views

Pole expansion and Fourier series of lemniscate sine function

Given that $$\frac{\varpi}{\text{sl}(\varpi z)}=\sum_{n,k\in\mathbb{Z}}\frac{(-1)^{n+k}}{z+n+ik}$$ It can be deduced that, for $-1<\text{Im}(z)<1$: $$\frac{1}{\text{sl}(\varpi z)}=\frac{\pi}{\...
Dqrksun's user avatar
  • 502
2 votes
2 answers
98 views

How can I find the periods of difference of the sine waves that has irrational coefficients

I'm working on a project right now. And now I need to find periods of difference of the sine waves and i'm stuck. In few resources I found that I can find the periods of summed or differenced sine ...
Eren Gümüş's user avatar
0 votes
1 answer
42 views

$2\pi$ periodic functions with null barycenter [closed]

Let $f$ be a continuous $2\pi$-periodic (real-valued) function satisfaying: $$\forall x \in [0, 2\pi),\ \forall n \ge 2,\ \sum_{k =1}^n f\left(x + \frac{2\pi}{n}k\right) = 0.$$ Can we conclude that ...
Tasty's user avatar
  • 102
2 votes
0 answers
54 views

$F(u,v)$ is a rational function. If $\pi$ is a period of $F(\cos x,\sin x)$, then $F(u,v)=F(-u,-v)$. "Introduction to Analysis" by Teiji Takagi.

I am reading "Introduction to Analysis" by Teiji Takagi. The author wrote the following proposition without a proof. Let $F(u,v)$ be a rational function. If $\pi$ is a period of $F(\cos x,\...
佐武五郎's user avatar
0 votes
1 answer
47 views

Finding the Fourier Coefficients for a piecewise function of sin

I’m having a great deal of trouble finding the Fourier Coefficients of the following function: $$f(x)=\begin{cases}0, & x\leq 0 \\ \sin x, & x>0\end{cases}$$ where $x$ is defined over the ...
H Franklin's user avatar
0 votes
1 answer
56 views

Finding the period of the BBS sequence

Let $n=pq$, where $p,q$ are primes and $p \equiv q \equiv 3 \mod 4$. Choose an integer, $x_0$, such that $x_0$ and $n$ are co-primes. We define the sequence: \begin{align} x_i = x_0^{2^i} \mod n \end{...
Giorgos Mitropoulos's user avatar
-1 votes
2 answers
49 views

If a periodic function is written as linear combination of 2 functions then must they also be periodic?

Given a non constant periodic function with period $T$, say $f(x)$. Also given that $f(x)$ can be written as a linear combination of 2 independent functions as $$f(x) = a \, g(x) + b \, h(x),$$ where $...
Artemis 's user avatar
4 votes
2 answers
142 views

Proving that a complicated complex expotential is periodic or not.

I'm trying to prove that $x[n]=\exp \left(j\left(\frac{\pi}{24} n^2+\frac{\pi}{36} n^3\right)\right), n \in \mathbb{Z}$ is not periodic. I thought of proving that there is no $N \in \mathbb{Z}$ such ...
Nyquist-er's user avatar
0 votes
0 answers
69 views

pdf transformation of periodic scalar random variables related by a derivative

Two random scalar variables, x and y, are related by the following expression: $y(t) = f[x(t)] = a\frac{dx(t)}{dt}$, where $x(t)$ is periodic and $a$ is a constant. How can I calculate the pdf $g_{Y}[...
unkown's user avatar
  • 11
1 vote
0 answers
106 views

Fourier transform of periodic distributions & local application of a result

Context: I am currently working through Chapter $8$ of Anders Vretblad's Fourier Analysis and Its Applications. This particular chapter focuses on distributions, and builds up to the Fourier transform ...
PrincessEev's user avatar
  • 44.4k
1 vote
0 answers
38 views

Proof that you can solve the discrete logarithm problem given the period of a certain function.

Given $q$ a prime number , $a$ a primitive root modulo $q$ and $b=a^x \pmod q$. The discrete logarithm problem is to find $x$ (specifically the smallest positive integer $x$ for which the previous ...
Omeglac's user avatar
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