0
votes
1answer
40 views

Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
0
votes
0answers
9 views

analytic property of periodic properties

Can any one suggest me some books in which I can see the analysis of periodic functions? I dont have any constrain in the domain or codomain. For example this book ...
3
votes
1answer
84 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 ...
4
votes
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 ...
0
votes
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, ...
1
vote
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 ...
4
votes
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 ...
6
votes
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 $f=g$
1
vote
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 ...
1
vote
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
votes
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
votes
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: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
1
vote
4answers
331 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 ...
2
votes
1answer
124 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 ...
1
vote
2answers
1k 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
0answers
105 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 ...
1
vote
0answers
102 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
85 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
96 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 ...
3
votes
1answer
203 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 ...
2
votes
2answers
268 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 ...