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

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2
votes
2answers
26 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
3
votes
1answer
109 views

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous periodic function with period $T>0$

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous periodic function with period $T>0$. Show that there exists an element $x \in \mathbb{R}$ for which $f(x)=f(x+T/2)$. This is what I came up with ...
0
votes
1answer
18 views

What is the period of $f(X) = (A+K\cdot X) \mod N$

I have a function $f$ mapping from integers to integers as follows: $f(X) = (A+K \cdot X) \mod N$ Where $A,K,X,N$ are positive integers. What is the period of the function?
2
votes
2answers
144 views

Fourier Series and periodicity

Let $f$ be a $2 \pi$-periodic piecewise continuous function and let \begin{equation} f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx} \right] \tag{*} \end{equation} ...
1
vote
0answers
26 views

Solutions for ODE with periodic B.C.

I have the following ODE $$-u(x)''=f(x) \qquad u:[0,1)$$ It is smooth and 1-periodic. Assume that I have a solution u(x). How do I prove that: $$u(x)= u(x)+c \quad c \in \mathbb{R} $$ is also a ...
1
vote
0answers
56 views

Triangulation in periodically repeated 3D box

I have a set of points in a 3D box (rectangular parallelepiped), which is periodically repeated out of two its opposite sides. How do I find the Delaunay triangulation for this set of points? Here is ...
1
vote
0answers
26 views

Fourier transform of $f[n]$ and $f[-n]$

Hi I am just wondering, If I have a signal $f[n]\in \mathbb{C}^L$, i.e. $f$ is $L$-periodic, i can also define $h[n]=f[-n]$. Is it true that the Fourier transform of $f$, say $\hat{F}$, and the ...
2
votes
2answers
556 views

Extend a function as odd/even periodic function

Let $f$ be the function $f(x) = x^2 + 2 $, where $ 0<x<1 $. Extend the function $f(x)$ (1) As an odd periodic function with period $2$ (2) As an even periodic function with period $2$ (3) ...
1
vote
1answer
61 views

How do I prove that $\cos(\frac{1}{2}x)$ is a periodic function?

Given: $f(x)=\cos(\frac{1}{2}x)$. Prove: f is a periodic function with period 4π My math teacher never went over this so I don't know where to start or what to do :/
1
vote
1answer
91 views

The difference between two periodic functions converges to zero, is this two functions identical?

If $f(x)$ and $g(x)$ are two periodic functions, that is, $f(x+T_1)=f(x)$ and $g(x+T_2)=g(x)$ for every $x \in R$. Now that $lim_{x\to\infty}(f(x)-g(x))=0$. Conjecture: $f(x) \equiv g(x)$.
0
votes
1answer
52 views

Simple question about complex $e^{i}$ and angles

I'm working with angles. I have a hard time figuring something. In electric physics, I have an equation describing an AC voltage function, this way $V_{x} = 0.0469 \cdot e^{-j \cdot 1.083}\cdot ...
0
votes
0answers
31 views

problem about Period

we have a, b, m initially where 0 <= a, b < m <= 10^7. And we define a[1] = a, ...
1
vote
1answer
204 views

fundamental period

$f:\Bbb R\to \Bbb R$ is continous and nonconstant function. Let $p$ be a positive real number such that $f(x+p)=f(x) $ for all $x \in \Bbb R$ . Then there exitst $n\in\Bbb N$ such that ${p \over n ...
1
vote
1answer
28 views

What functions model this relationship?

I'm currently working a bit on an AI, and in order for it to function, it must be able to quickly predict where a point will be in space, given any distance. The movement of this point may be modeled ...
4
votes
1answer
95 views

$f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ nonconstant, continuous, with period $1, \sqrt{2}$, respectively, then $f_1 + f_2$ is not periodic

I've been working on this problem for several hours, but I keep getting stuck. Suppose $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ periodic with period $1, \sqrt{2}$, respectively, and that each of ...
1
vote
1answer
169 views

Prove the included formula relating cos(nx) and cos(x)

I'm struggling with the below problem. Can anyone shed some light on it? Show that the below formula is a correct relation between $y = \cos n\theta$ and $x = cos \theta$ for all $n$: $$ x = \frac 12 ...
-2
votes
1answer
133 views

Is this function periodic? [closed]

Is the following function periodic? $$f(x)=\cos(x)*\cos(x\sqrt5)$$ A function $f$ is said to be periodic with period $P$ ($P$ being a nonzero constant) if we have $$f(x+P) = f(x)$$ for all ...
3
votes
1answer
213 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and ...
0
votes
1answer
55 views

Periodic extensions and continuity

Consider the periodic extension $g$ of $f(x)=x^2-1$ where $x \in (0,1)$. Now suppose that $h(x)$ is a continuous function, does $h$ composed with $g$ have to be continuous? I don't think it should, ...
1
vote
1answer
47 views

Periodicity of a function

If f(x) is periodic with period a, would f(tx) be periodic with period a/t? Would f(tx+b) make still have period a/t? Im inclined to think so, because this works for the trig functions, but i'm not ...
1
vote
1answer
127 views

Understanding a periodic discontinuous function

This is from an example in my PDE text. It's something I should probably know, but maybe I am just reading the wording wrong. The text (Asmar as it happens, section 2.1) gives the following example ...
1
vote
1answer
117 views

Linear fit to wrapped / periodic data

Assume I have a number $N$ of noisy (phase) data $\varphi_i$, i.e., $\varphi_i \in [0,2\pi)$. I know that between $\varphi_{i-1}$ and $\varphi_i$ there is a constant phase shift $\Delta \varphi$ ...
1
vote
1answer
81 views

Period of function $x\mapsto (f(x))^2$

If i know, that $f(x)$ is a periodic function with period $T$, how should i prove, that $(f(x))^2$ has period $T_1 : T_1 \le T$. I tried to use periodic function determination: $f(x)=f(x+T)$, but it ...
2
votes
1answer
441 views

How to prove a function is periodic?

$$f(x)=\begin{cases}1&\text{if }2n-1<x<2n,\\0&\text{if }2n<x<2n+1. \end{cases} $$ Is this function is periodic or not? How can I prove it?
0
votes
1answer
26 views

Periodicity of a series of points

I have a series of 2-D points. I want to analyze if there is any pattern in this series. Also, if the series is somehow repeating, can I extract a smaller series of points to analyze instead of the ...
5
votes
2answers
164 views

If $f$ is a continuous periodic function with irrational period and if $\sum_n\frac{|f(n)|}{n}<\infty$, then $f$ is identically zero.

Please show that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous periodic function with irrational period and if $\sum_n\frac{|f(n)|}{n}<\infty$, then $f$ is identically zero. (For example, using ...
2
votes
2answers
310 views

How to make ANY function to repeat periodically?

I have some function, say, Gauss PDF Now I want it to repeat, say, every N units How to transform any function this way? I know I can sum function at each shift. But is it possible to convert ...
2
votes
0answers
110 views

Periodic Laplace transform

Here's the questions and the graph I've been struggling with this since Thursday and this is due today. I need help on problems a and b. For a, the question is: "Write $f(t) = \sum_{n=0}^\infty ...
5
votes
2answers
155 views

For $f$ periodic, $g\to 0$ the integral of $fg$ converges (under some more conditions)

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions such that: $f$ is periodic, with finitely many zeros in a period The average value of $f$ on a period is $0$ $g$ is monotonic decreasing and ...
6
votes
4answers
293 views

Prove that f(x)=g(x)

Show that if $f,g:\mathbb{R}\to \mathbb{R}$ are continuous and periodic and $\lim_{x\to \infty}[f(x)-g(x)]=0$, then $f=g$
1
vote
2answers
83 views

An inequality on $C^1$ periodic functions

Suppose $f \in C^1(\mathbb{R})$ and $f(x + 1) = f(x) \ \forall x \in \mathbb{R}$. Show that $$||f||_{\infty} \leq \int_0^1|f| + \int_0^1|f'|.$$ I have tried using techniques in Fourier Analysis such ...
1
vote
1answer
46 views

Name for points with same value in a periodic function

I'm looking for a name to describe the points of the same value of a periodic function -- you draw a horizontal line across a periodic function and you get a set of values. That is, a name for _all ...
1
vote
0answers
78 views

Is there a generic approach to Generating Function of periodic sequences?

Recently I read on wiki (see here): "Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones." ...
0
votes
0answers
45 views

Finding The Periodicity of $f(x) = \sin(x/2)\cos(x/3)$ [duplicate]

I want to find the periodicity of the following function: $$f(x) = \sin(x/2)\cos(x/3).$$ I have calculated the periodicity of the above functions which is $\pi/6$. Is it correct? Can you please help ...
0
votes
0answers
372 views

Find the period of sequence.

A sequence is such that its terms are generated by the formula: $$r_i =(ar_{i-1}+b\pmod m)$$ where $a,b,m,r_0$ are given. find the period,that is the number of terms that are repeated. For example, ...
1
vote
1answer
86 views

Exercice on periodic function

Let $f$ be a periodic function, $\mathcal{C}^1$ on $\mathbb{R}$ such that: $$\displaystyle\int_0^{2 \pi} f(t) \, dt = 0$$ $$f(2 \pi) = f(0)$$ Prove that $$\forall t \in [0,2 \pi]: \int_0^{2 \pi} ...
3
votes
2answers
147 views

Period of derivative is the period of the original function

Let $f:I\to\mathbb R$ be a differentiable and periodic function with prime/minimum period $T$ (it is $T$-periodic) that is, $f(x+T) = f(x)$ for all $x\in I$. It is clear that $$ f'(x) = \lim_{h\to ...
3
votes
1answer
108 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
711 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
364 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
430 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
346 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
439 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] ...