Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.
1
vote
2answers
489 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
536 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
193 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
82 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
157 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 ...
4
votes
2answers
286 views
Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?
I have been looking at concise ways to represent irrational numbers using only integers.
I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...
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
1answer
79 views
Antiperiodic functions are indecomposable
Let $A$ be the algebra of continuous functions $\mathbb{R} \to \mathbb{R}$ which are periodic of period $1$, and write $M$ for the $A$-module of antiperiodic functions of period $1$ (meaning $f \in M$ ...
0
votes
1answer
95 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 ...
4
votes
1answer
2k views
Period of the sum/product of two functions
Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link ...
0
votes
0answers
60 views
how can I make these equations as periodic oscillation?
I am trying to implement the Predator-Prey System. And I need to find the constants which yield a stable oscillation.
here is the equations:
$$
x(i + 1) = x(i) + a x(i) + b y(i) + e x(i) x(i)
$$
$$
...
0
votes
1answer
161 views
Find the period of the following function
suppose that we have function
$y=[2x]-3*[4x]$
here $[*]$ denotes as a minimum distance till integer.
we are required to find period of this function,first of all i am confused in terms of ...
1
vote
2answers
66 views
a Function with several periods
A periodic function is given by $ f(x+nT)=f(x) $, with 'n' an integer and T the period.
My question is if we can define a non-constant function with several periods; by that, I mean
$ ...
1
vote
0answers
28 views
Asymptotic order of some sums with the Fourier coefficients
Given $f\in C^{w}[0,1]$ with periodic conditions $f(0)^{(j)}=f(1)^{(j)},\ j=0,\dots, w-1$ and its Fourier series are $f(x)=\sum_{l}f_{i}\exp(2\pi ix)$.
I need to find the asymptotic order of errors ...
0
votes
1answer
140 views
Fourier Coefficients of periodic function
Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$
known?
There are a lot ...
1
vote
0answers
71 views
Non-periodic BV function
I want to know the definition of non-periodic bounded variation function.
I know the definition for periodic function of bounded variation, which is,
Let $f:[a,b]\to \mathcal c$ and ...
2
votes
1answer
77 views
To show $\int_{-\pi+x}^{\pi+x}f(u) du=\int_{-\pi}^{\pi}f(u) du$. [duplicate]
Possible Duplicate:
An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
I have to show if $f\in \mathcal L[-\pi,\pi]$ and if $$f(u+2\pi)=f(u), ...
2
votes
4answers
85 views
Variation of periodic function
If $f\colon\Bbb R\to\Bbb R$ then $\operatorname{var}(f, [a,b]):=\sup \{\sum_{k=1}^n |f(x_k)-f(x_{k-1})| \}$,
where supremum is taken over all finite sequences $(x_k)$ such that ...
1
vote
1answer
166 views
How to effectively compute a periodic function?
I'm writing a program to compute a value of periodic function for any arbitrary large argument:
$f(k) = (\sum_{n=1}^{2^k} n)\mod\ (10^9 + 7)$, where $n,k \in \mathbb{N} $
I know that $ f(k + P) = ...
1
vote
1answer
107 views
How to construct and oscillation with exponentially growing period times?
I'm searching for the (maybe even smooth) "oscillating" function
$$f(t)=A\sin{\left(g(t)\right)},$$
such that there are zeroes at times $t_n=T^n$ for some fixed number $T$. So this will not really ...
4
votes
2answers
361 views
What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?
So, I have a general question first.
What happens to the periodicity when we multiply two periodic trig functions with one another ?
The next one is very specific, what is the period of the function ...
2
votes
1answer
96 views
How to determine periodicity of complex log in different bases?
How do you determine the "period" of a complex logarithm as a multivalued function in an arbitrary (real or complex) base?
I apologize in advance if my terminology is incorrect, but let me illustrate ...
2
votes
2answers
188 views
Limit of integral involving periodic function
Can anyone help me with this? I want to know how to solve it.
Let $f:\mathbb R \longrightarrow \mathbb R$ be a continuous function with period $P$. Also suppose that ...
1
vote
1answer
262 views
What's the prime period of $\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$
What's the prime period of the following function?
$$\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$$
4
votes
1answer
114 views
Triangular periodic tessellation in two variables
The simple function $f(x, y) = \cos(x) + \cos(y)$ is doubly periodic in a square tessellation. One feature of this surface is that cuts through the nearest extrema in either of two directions (e.g. ...
0
votes
2answers
35 views
Is it okay to use the same variable to describe function periods?
If I have a system of period functions of $x$ and $y$, in this case trigonometric,
$$\begin{cases}
\sin{(2x + y)} = 0 \\
\sin{(2y + x)} = 0
\end{cases}$$
is it okay for me to to use the ...
2
votes
1answer
410 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 ...
2
votes
3answers
387 views
Does the phrase “instantaneous frequency” make sense?
I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a ...
0
votes
1answer
73 views
Function with oscillating frequency?
I'm looking for a function whose frequency oscillates around a certain value (say, oscillating between 440 Hz and 880 Hz, at a rate of 1 Hz -- i.e., its frequency goes up and down once per second, ...
3
votes
4answers
677 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 ...
