# 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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### Why is $|\cos(x)|$ piecewise differentiable on $x \in [-\pi, \pi)$?

I am asked if the $2\pi$-periodic function $f(x) = |\cos(x)|$ is piecewise differentiable on the interval $x \in [-\pi, \pi)$. Plotting the function yields this graph, which shows that there are ...
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### Weierstrass elliptic function for quaternions?

What if we generalize and modify the Weierstrass elliptic function for quaternions ? So our function could have $2,3$ or $4$ periods. How would that theory be like ? Is matrix representation the key ...
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### Why is this function periodic?

f(x) = {1 if x is rational, 0 if x is irrational} Why is this function periodic? I tried the following to prove it but couldn't find it satisfying. What I did was assume that f(x+T) = f(x), with T a ...
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### Preperiod and Period of the nim-sequence of Octal Games .17 and .117

This refers to a type of impartial game defined as octal games by Berlekamp, Guy and Conway in the first edition of the Winning Ways books. I noticed that the nim-sequences of $.17$ and $.117$ (first ...
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### Any bounded orbit converges to a unique critical point if a system has no periodic orbit

If a system in $\mathbb{R}^2$ has no periodic orbit, and has only one critical point which is asymptotically stable, prove that any bounded orbit $(x(t):t>0)$ must converge to the unique critical ...
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### Period of $\sin^{2} x\ + \cos^{2} x$

While I was studying Periodicity of functions I encountered the following problem : My textbook said that : Constant function is a periodic function but its fundamental period doesn't exist So to ...
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### Entire function taking real values on two parallel lines is periodic [duplicate]

Given an entire function $f$, I have that $f(ix)$ and $f(1+ix)$ are real for all $x \in \mathbb{R}$. I want to show that $f(z) = f(z+2)$ for all $z \in \mathbb{Z}$. I've thought about considering the ...
1 vote
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### Compute $\lim_n \|u_n-\bar f\|_p$

Let $p \in [1, \infty]$. Let $f \in L^p_{\text{loc}} (\mathbb R)$ be $T$-periodic, i.e., $f(x+T) = f(x)$ a.e. $x \in \mathbb R$. Let $$\bar f := \frac{1}{T} \int_0^T f (t) \, dt.$$ We define a ...
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### $u_n \to \overline f$ in the weak topology $\sigma(L^p, L^{p'})$

Let $p \in [1, \infty]$. Let $f \in L^p_{\text{loc}} (\mathbb R)$ be $T$-periodic, i.e., $f(x+T) = f(x)$ a.e. $x \in \mathbb R$. Let $$\overline f := \frac{1}{T} \int_0^T f (t) \, dt.$$ We define a ...
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### $2L$ periodic Sobolev space

I am interested in studying the space $H^1([0,1])$. Using Fourier series, I am the. naturally led to studying the corresponding periodic Sobolev space. All references I can find on the internet deal ...
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### frequency and periodicity

This is really a basic question: Let's say we have a message that is sent 2 times each second (periodic message). Thus, we can say that each 0.5s we send a message. But I am trying to understand it in ...
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### Fundamental Period vs function continuous atleast once

I know that function is periodic, non constant and continuous at least once $\implies$ function has a fundamental period My question: Can we construct a function which is discontinuous everywhere and ...
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### Counter-example of continuous Z-periodic functions

In Chapter 16 of Tao's Analysis II, while giving the definition of $C(\mathbb{R/Z},\mathbb{C})$, Tao also has add the following one-sentence claim: By "continuous" we mean continuous at all ...
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### Approximation of the Bernoulli periodic function

I remember seeing a paper that provided a summation approximation of the Bernoulli periodic function which converges when $p\ge 2$; $$\dfrac{P_{p}(x)}{(p!)}$$ but I don’t quite remember it, I know for ...