Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.
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1answer
133 views
Fourier Coefficients of periodic function
Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$
known?
There are a lot ...
5
votes
5answers
1k 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?
6
votes
2answers
158 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 ...
7
votes
2answers
144 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 ...
1
vote
0answers
26 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
67 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 ...
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votes
0answers
42 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
66 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
1answer
51 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 ...
1
vote
2answers
40 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 ...
5
votes
3answers
157 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
41 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
76 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
52 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)?
0
votes
0answers
44 views
Negative periodic functions
Can any one tell me how to find a periodic function $f(t)<-k$, $k$ is strictly positive constant, but I don't need functions where we just add a negative number to an usual periodic function like:
...
3
votes
2answers
76 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)} ...
3
votes
3answers
214 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
80 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
33 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
42 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
143 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
0answers
45 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 ...
1
vote
0answers
24 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 ...
2
votes
2answers
86 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 ...
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votes
1answer
50 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, ...
5
votes
2answers
226 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
57 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
64 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 ...
3
votes
3answers
142 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.
1
vote
2answers
59 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
40 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
133 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 ...
3
votes
4answers
646 views
Are sin and cos the only continuous and infinitely differentiable periodic functions we have?
Sin and cos are everywhere continuous and infinitely differentiable. Those are nice properties to have. They come from the unit circle.
It seems there's no other periodic function that is also ...
1
vote
1answer
29 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
39 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, ...
0
votes
0answers
50 views
Set of N Periodic & Non Negative Integer Sequences forming by summation, the series of Whole Numbers.
I'm interested in Set of N non negative and periodic/oscillating integer sequences,
that once summed forme the serie of whole numbers: 0,1,2,3,4,5,...,K
For exemple:
With N = 2 (A,B), K = 7 and ...
2
votes
0answers
101 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
254 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 ...
4
votes
1answer
187 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\}$
...
0
votes
3answers
88 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?
7
votes
2answers
131 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
84 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 ...
1
vote
0answers
94 views
sum of periodic function which eventually vanish
It is known that : If $\lim_{x \to \infty} \sum_{i=1}^n f_i(x)=0$ and $f_i$ are real-valued periodic function, then $\sum_{i=1}^n f_i(x)=0$ for all $x \in \mathbb{R}$. We can solve this problem by ...
3
votes
0answers
262 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$$
there is a very practical way to evaluate it by interpreting it as some ...
2
votes
1answer
62 views
Relation on fourier coefficients implies smoothness for a periodic continuous function
I just came across with the following question.. suppose we are given a periodic function of period $2\pi$. We define $a_n$ and $b_n$ to be the Fourier coefficients of $f$. To be precise, we have ...
8
votes
3answers
131 views
Integral inequality on a periodic function
Given $f:\mathbb R\to \mathbb R^+$ continuous and periodic of period $T\geq 0$, I am asked to prove that $$\int_0^T\frac{f(x)}{f(x+\alpha)}\mathrm dx\geq T,\; \forall \alpha\in\mathbb R.$$ How does ...
2
votes
1answer
398 views
FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions
In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ created by a charge distribution $\rho(\mathbf{r})$ is
$$ \Delta ...
0
votes
1answer
94 views
How can a Bézier curve be periodic?
As I know it, a periodic function is a function that repeats its values in regular intervals or period. However Bézier curves can also be periodic which means closed as opposed to non-periodic which ...
1
vote
2answers
464 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
votes
2answers
506 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 ...
