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
147 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 ...
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1answer
25 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
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1answer
93 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 ...
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1answer
126 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 ...
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1answer
123 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
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1answer
198 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 ...
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1answer
43 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, ...
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1answer
40 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 ...
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1answer
90 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 ...
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1answer
79 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 $\Delta \varphi$ ...
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1answer
63 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
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1answer
337 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?
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1answer
21 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 ...
4
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2answers
147 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 ...
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2answers
197 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
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0answers
100 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 ...
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0answers
105 views

Piecewise form of a triangle function

Find $f(t) = \sum U(t-a)f(t-a)$ For a triangle wave functions with $(0,1)$ $(\pi,-1)$ $(2\pi,1)$, $(3\pi,-1)$ ... as its points. With a period of $2\pi$. So I found that the first segment from ...
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2answers
145 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
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4answers
276 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 $f=g$
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2answers
77 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 ...
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1answer
45 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 ...
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0answers
70 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." ...
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0answers
38 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 ...
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0answers
264 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, ...
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1answer
77 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
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2answers
107 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
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1answer
99 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 ...
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1answer
116 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 ...
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1answer
24 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 ...
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0answers
66 views

A general theorem for periodic sequences?

I wonder whether from a very formal point of view one could regard with certainty the following statement on periodic sequences (1) as directly proven through Fourier's theorem (ref. inverse discrete ...
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2answers
541 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 ...
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2answers
338 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 ...
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0answers
116 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: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
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50 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 $1<q<\infty$. ...
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4answers
330 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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1answer
272 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 ...
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0answers
381 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 ...
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2answers
68 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 ...
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3answers
738 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 ...
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0answers
59 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] ...
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3answers
261 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: ...
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2answers
274 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
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2answers
163 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)} ...
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1answer
143 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 ...
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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 ...
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3answers
256 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. ...
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2answers
63 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) ...
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1answer
67 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 ...
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3answers
473 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
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1answer
71 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 ...