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

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3
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1answer
109 views

Integral inequality using positive and negative parts

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable (with respect to the Lebesgue measure), $\pi$-periodic function which is Lebesgue integrable over $[0,\pi]$. Moreover assume that ...
2
votes
1answer
119 views

How to show $C_p^k([-\ell, \ell])$ is not a Banach space?

I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms ...
0
votes
1answer
26 views

Convergence of a series of a given metric..

I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow ...
11
votes
2answers
770 views

$\cos(x)+\cos(x\sqrt{2})$ is not periodic

Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
6
votes
2answers
372 views

How to expand the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} $?

My Question: My Goal is to determine the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} \quad$ for $x \in [-\pi, \pi ]$ This function is $2\pi$-periodic. My Approach: i found ...
3
votes
0answers
124 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
2
votes
0answers
53 views

Does weak convergence (of a periodic function) imply weak-* convergence for the derivative?

Lets assume we have $b(x,t) \in L^\infty$ periodic in x and $\frac{db}{dt} \in L^{p}, p\in\mathbb{N}$. $b(x,t)$ converges weakly to an $f(t)=\int b(x,t)dx \in L^q$ for all $1<q<\infty$. ...
1
vote
4answers
473 views

Boundedness and Uniform Continuity

Prove that a continuous periodic function on $\mathbb{R}$ is bounded and uniformly continuous on $\mathbb{R}$. Given the continuous periodic function $f:\mathbb{R} \to \mathbb{R}$ for some period ...
3
votes
2answers
363 views

Laplace transform of a periodic function

Knowing that $$L[f(t)]=\frac1{1-e^{-sp}}\int_0^{p} e^{-st}f(t)dt$$ $p$ indicates the period of the function If $f$ is a continuous function by segments in $[0,\infty)$ and $F(s)=L[f(t)]$ exists for ...
1
vote
0answers
481 views

Solving a Forced Oscillations Differential Equation Problem

A building consists of two floors. The fi rst floor is attached rigidly to the ground, and the second floor is of mass m = 1000 slugs (fps units) and weighs 16 tons (32,000 lb). The elastic frame of ...
1
vote
2answers
79 views

Set of periods of a real-valued function

Let $~f : E \rightarrow \mathbb{R}, ~ E \subset \mathbb{R}~~$ be a periodic fucntion and $ S = \{ T \in \mathbb{R} ~ : ~ \forall x\in \mathbb{R} ~~ f(x+T) = f(x) \} $ be the set of all periods of ...
6
votes
3answers
1k views

Is a broken clock right twice a day?

I was thinking about this expression, and I wondered if it holds true when the clock is slow. I can imagine a slow clock which is not right at all in the span of 12 hours—imagine a clock which ticks 5 ...
4
votes
0answers
64 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
3
votes
3answers
375 views

Periodic polynomial?

I was thinking if it was possible to create a polynomial that would be periodic all over the reals, since polynomials can be periodic on an interval. I then I found out the following function: ...
0
votes
2answers
424 views

Trigonometric Identities - Assignment

How do I simplify Cos(5 theta)? I got as far as Cos(2theta + 3theta). Do I then say Cos(2theta + 3theta) = Cos(2theta) + Cos(3theta)? In that case, how do I get Cos(3theta)?
3
votes
2answers
196 views

Limit of an integral with a periodic function

Let $f , g$ be continuous functions: $f:[0 , 2\pi]\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$. Assume $\forall x\in\mathbb{R}:g(x+2\pi)=g(x)$ and $$\int\limits_{0}^{2\pi} \! {g(x)} ...
1
vote
1answer
162 views

Type of periodicity in champernowne constant.

Digits of Champernowne constant are aperiodic, else it will be rational. Fine! But it is not random because I can write a program which will give me the position of every digit. E.g. I can calculate ...
3
votes
3answers
1k views

Express $\sin 4\theta$ by formulae involving $\sin$ and $\cos$ and its powers.

I have an assignment question that says "Express $\sin 4\theta$ by formulae involving $\sin$ and $\cos$ and its powers." I'm told that $\sin 2\theta = 2 \sin\theta \cos\theta$ but I don't know how ...
1
vote
3answers
335 views

Checking whether a function is even or odd and checking if a function is periodic

For given function, for example $f(x)=x^3+x^2-x-1$, to check whether it's even or odd, we have to find $f(-x)$. Therefore, $f(-x)=-x^3+x^2+x-1$, which proves the function is not odd neither even. ...
0
votes
2answers
82 views

Find the period of the related function

If F is any function with a period of $6$, determine the period of each related function below: $y = f(x+1)$ $\displaystyle y = f(\frac{x}{2})$ I know that the basic definition of a period is $f(x) ...
0
votes
1answer
71 views

Cleaning a signal and computing period

I am working with a signal which is a periodic square signal with some kind of noise and some outliers. I would like to know which is the best solution in order to get the period and clean the ...
2
votes
3answers
667 views

How to find the period of a periodic function?

I'm working on some numerical analysis problem, and I'm studying functions that "seem" to be periodic. Now what I would like to do, is to determine their period. Only, the methods I actually use are ...
0
votes
1answer
76 views

how can I tell the difference bewteen chaos and periodic with lots of noise?

I am most interested in the difference of power spectrums between chaos and periodicity. I read some paper on testing chaos against some more general alternatives (mostly stochastic systems) through ...
2
votes
0answers
56 views

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
3
votes
2answers
218 views

Show that a the periodic function is even in a specific interval

I have just started to learn about Fourier series, and Even/Odd functions. I am supposed to show that the function below is even in the given period. I assumed that if I tried solving the $B_n$ it ...
-1
votes
1answer
81 views

Prove the orbits of ODE system to be periodic?

For the parallel flow $\dot{\theta_1}=w_1$, $\dot{\theta_2}=w_2$ on the 2-torus, where $w_1$ and $w_2$ are positive and where the coordinates $\theta_1$ and $\theta_2$ are taken modulo 1. Also, ...
7
votes
2answers
951 views

Is $f(x)=\sin(x^2)$ periodic?

Is the function $f:\Bbb R \rightarrow \Bbb R$ defined as $f(x)=\sin(x^2)$, for all $x\in\Bbb R$, periodic? Here's my attempt to solve this: Let's assume that it is periodic. For a function to ...
1
vote
2answers
178 views

Asymptotic Expansion of a Multiscale Partial Differential Equation

I'm trying to understand how to solve $$-\nabla\cdot(K(\frac{x}{\epsilon})\nabla u(x,\frac{x}{\epsilon})=f \text{ in } \Omega$$ $$u(x,\frac{x}{\epsilon})=u_D \text{ in } \partial \Omega$$ where ...
1
vote
0answers
111 views

Inequalities of integrals of periodic functions

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
6
votes
7answers
11k views

How to find period of this periodic function?

How can I find a period of this function? $$2\sin{3x} + 3\sin{2x}$$ Is here any way how to sum both sinuses?
3
votes
3answers
165 views

Does anyone know of any additive periodic functions?

Anyone know of any periodic functions satisfying $f(xy)=f(x)+f(y)$, when gcd(x,y)=1, I need a function other then the function $a_d(k)$, thats 1 if d divides k, and 0 if it doesn't.
2
votes
2answers
114 views

calculate the period of an hypotrochoid

I'm curious how to find out the period of an hypotrochoid. x = (a-b) * cos(t) + h * cos( ((a-b)/b) * t ) y = (a-b) * sin(t) - h * sin( ((a-b)/b) * t ) I know ...
0
votes
0answers
68 views

Factoring from order finding?

It appears that if you have a finite field, F_p, and you can factor the order of the field, p-1, then you can easily construct an arbitrary nth root of unity in F_p and also check that it be primitive ...
6
votes
3answers
224 views

prove a function to be constant

Let the function $f\colon \mathbb{R} \to\mathbb{R}$ have arbitrarily small positive periods, in the sense that if $\delta>0$ then there exists $T \in (0,\delta)$ such that $f(t + T) = f(t)$ for all ...
1
vote
1answer
45 views

How to prove the periodicity of an iterated function?

How to prove the periodicity of an iterated function? For example, how to prove $\sin_{[n]}(x)$ is a periodic function of period $2\pi$ $\forall n\in\mathbb{R}^+\cap\{0\}$ ?
0
votes
1answer
88 views

For the sum to be periodic, $f_1$ and $f_2$ must be commensurable;

For the sum to be periodic, $f_1$ and $f_2$ must be commensurable; that is, there most be a number $f_0$ contained in each an integral number of times. Thus, if $f_0$ is the largest such number, ...
2
votes
0answers
250 views

Fourier Series problem

Suppose you are given the following information about a continuous-time periodic signal, $x(t)$, with period $6$ and its Fourier series coefficients $(a_k)$, (1)-(4). Using the synthesis equation, ...
2
votes
2answers
680 views

Finding additional function values of an odd-periodic function.

I'm in a calc I class where I'm faced with the question: Suppose that f(x) is an odd function, and periodic with period 10. If f(3) = 4, find f(7) + f(5). Unfortunately, this is not talked about ...
0
votes
3answers
100 views

Is $y(t) = t^2 + i\cdot t^2$ a periodic function?

Is complex valued function like $y(t) = t^2 + i\cdot t^2$ a periodic function?
8
votes
2answers
163 views

Are there functions that have $\Re(f(z))$ periodic but $f(z)$ is not periodic?

Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number. Are there such ...
5
votes
1answer
100 views

Functions whose $n^{th}$ Derivatives form a cycle

The simplest example of this is $e^x$ which we could say has period 1 (it is its own derivative). $e^{-x}$ would have period 2. Using similar constructions, I can get a function that has a derivative ...
3
votes
0answers
534 views

Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
2
votes
1answer
144 views

Relation on fourier coefficients implies smoothness for a periodic continuous function

I just came across with the following question.. suppose we are given a periodic function of period $2\pi$. We define $a_n$ and $b_n$ to be the Fourier coefficients of $f$. To be precise, we have ...
8
votes
3answers
193 views

Integral inequality on a periodic function

Given $f:\mathbb R\to \mathbb R^+$ continuous and periodic of period $T\geq 0$, I am asked to prove that $$\int_0^T\frac{f(x)}{f(x+\alpha)}\mathrm dx\geq T,\; \forall \alpha\in\mathbb R.$$ How does ...
5
votes
1answer
361 views

Elliptic functions and (meromorphic) simply periodic functions.

Let be $f:\mathbb C\rightarrow\overline {\mathbb C}$ a meromorphic function. The set of periods $\Omega_f$ is a discrete (additive) group and we have one of these possibilities: i) $\Omega_f= \{0\}$ ...
2
votes
2answers
866 views

Periodic solution of differential equation

let be the ODE $ -y''(x)+f(x)y(x)=0 $ if the function $ f(x+T)=f(x) $ is PERIODIC does it mean that the ODE has only periodic solutions ? if all the solutions are periodic , then can all be ...
1
vote
2answers
2k views

Proof that a periodic function is bounded and uniformly continuous.

I need to show that if $f:\mathbb{R}\to \mathbb{R}$ is continuous and $\forall x \in \mathbb R, f(x+1)=f(x)$, then: $f$ is bounded, $f$ is uniformly continuous, there exists $c\in \mathbb{R}$ such ...
1
vote
3answers
805 views

When is the integral of a periodic function periodic?

I'm attempting some questions from Zwiebach - A First Course in String Theory, and have got stuck. I've proved that a function $h'(u)$ is periodic. The question then asks me to show that ...
1
vote
1answer
160 views

How can you find the 3d period of a summation of plane waves?

I realize this is a very hard question. In the very least I'd like to know if there is a way to do this or not. Say you have a summation of plane waves in a 3d volume, with longitudinal and ...
0
votes
2answers
429 views

Period of a finite binary sequence

Let $G:N\to\{0,1\}$, and let $L$ be some period of $G$, so that $G(i+kL)=G(i)$. What's the best a good way to find the smallest period of $G$? I mean an algorithm that takes ($G$,$L$) and outputs the ...