2
votes
2answers
64 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
4
votes
4answers
86 views

a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
2
votes
1answer
39 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
0
votes
0answers
82 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
1
vote
1answer
45 views

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$.

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$. I have computed $$c_n = \frac 1 {2\pi} \int^{\pi}_{-\pi} |\sin(x)|e^{-inx} dx = ...
2
votes
0answers
23 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
1
vote
1answer
22 views

Prove that $||f||_2 \le \sqrt{2 \pi} ||f || _{\infty}$

Let $||f||_2=\sqrt{\int_{-\pi}^{\pi} f^2(x) dx}$ $||f||_{\infty}=\sup \{ |f(x)| \mid x \in [-\pi,\pi]\}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ an in the space of piecewise continuous functions ...
1
vote
1answer
46 views

Uniform convergence of the Fourier Series using Bessel's inequality

Consider the Fourier series of $f$, $$ \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx) $$ Let $$f_n(x)= a_n \cos(nx) + b_n \sin(nx)$$ Then to show that $f_n(x)$ is uniformly ...
1
vote
0answers
83 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
1
vote
1answer
41 views

Degree of smoothness of real functions and Fourier series

I was wondering, why is the degree of smoothness $S$ of functions an integer? Why can't there be functions with $S=2/3$ ? The way we determine how smooth a function is by how many continuous ...
1
vote
2answers
21 views

Fourier coefficients assume a maximum and minimum?

Let $f:\mathbb R\to\mathbb R$ be continuously differentiable and periodic with period $2\pi$. The Fourier coefficients are defined by $$\hat f_n=\int_{-\pi}^\pi f(x)\exp(-inx)dx$$ My questions: Is ...
1
vote
0answers
12 views

fourier series of $f \circ R_\alpha$

I have a problem with the following: If $f \in L^2 (T)$ and $\displaystyle f(x)= \sum_{n=-\infty}^{\infty} a_n e^{2 \pi i nx } $ in $L^2(T)$ the Fourier expansion, then why the Fourier expand of $ f ...
4
votes
1answer
53 views

Leibniz series for $\pi$ using an integral of the Dirichlet kernel

I'm trying to create a proof of the Leibniz series $\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} = \frac{\pi}{4}$ using the Dirichlet Kernel. What I did is start with the kernel $$1+2 \left ( 1+\cos\theta ...
-1
votes
1answer
73 views

Is it true that the Fourier coefficient of convolution is the product of the coefficients?

what I mean by the title is the following: if we define the convolution between two $2\pi$-periodic, $C^1$ functions as $f*g(x) = (2\pi)^{-1}\int_{-\pi}^\pi f(x-y)g(y)dy$, is it true that the Fourier ...
3
votes
0answers
32 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
1
vote
0answers
34 views

The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
2
votes
1answer
88 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
3
votes
1answer
113 views

Convergence of series of functions: $f_n(x)=u_n\sin(nx)$

Let $f_n(x)=u_n\sin(nx)$ where $\displaystyle\sum f_n$ converges pointwise, and $ \displaystyle x \mapsto \sum_{n=0}^{+\infty} f_n(x)$ is continuous. Prove that $ u_n\rightarrow 0$ when n ...
1
vote
1answer
32 views

holomorphic function with integral coefficients

I'm trying to prove that an holomorphic function on $\{Z, |Z|<1\}$ and continuous on $\{Z, |Z|\leq 1\}$ with coefficients in $\mathbb Z$ is polynomial. I have tried to establish some partial ...
0
votes
1answer
41 views

Is continuous function space with standard inner product on $\big[0,\frac{1}{2}\big]$ not complete?

I think Fourier approximation on step function is one example of incompleteness, is it true? Or could you suggest any intuitive examples for incompleteness?
7
votes
4answers
190 views

Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

I am trying to solve exercises for the coming exam, and I am stuck on this exercise: Evaluate $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
1
vote
0answers
33 views

A question related to the Zygmund functions

Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, ...
1
vote
1answer
69 views

Decay of Fourier coefficients sequence

If $f:\Bbb R\to \Bbb R$ is a $2\pi-$ periodic, $C^1$ function, then $k^2a_{k}(f)\to 0$ where $$a_{k}(f)=\frac {1}{\pi}\int_{-\pi}^{\pi}f(x)\cos kx dx$$ are the Fourier coefficients. I ask if this is ...
1
vote
2answers
87 views

Let $f : [0;1] \to \mathbb{R}$ be a continuous function such that $f(0) = 0$. Which of the following statements are true?

Let $f : [0;1] \to \mathbb{R}$ be a continuous function such that $f(0) = 0$. Which of the following statements are true? a. If $\int_ 0^{\pi} f(t) \cos nt\, dt = 0,$ for all $n \in {0} \cup ...
8
votes
1answer
114 views

$\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt) \,dt$?

Let $f \in C[-\pi,\pi]$. Find the following limit: $$\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt)\,dt\,?$$
2
votes
0answers
32 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

Let $K$ be a fixed integer, and $\mathcal{F}$ the set of trigonometric polynomials with at most $K$ nonzero terms. Let $(f_n)$ be a sequence in $\mathcal{F}$ converging pointwise (on $\mathbb{R}$) to ...
1
vote
1answer
132 views

Find the sum of a series using Fourier series

Calculate the Fourier series of the function $f(x)=\sin{(ax)}$ on the interval $[-\pi,\pi]$ and then calculate: $$\sum_{n=1}^{\infty}{\frac{n^{2}}{(4n^2-1)^2}}$$ I've already calculated the ...
0
votes
1answer
221 views

Details for proof of Poisson summation formula

In the proof of the Poisson summation formula, there is a detail which is not clear to me how to resolve. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz-class function. Let ...
0
votes
1answer
34 views

uniform convergence of series and Fourier coeffient

let ${ \phi_{n} } $ be sequence of orthogonal functions on $[a,b]$ If the series $ \sum_{n=1}^{\infty } a_{n}\phi_{n}(x) $ converges uniformly to a function $f(x)$ on $[a,b]$ prove that for each $n ...
3
votes
1answer
59 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
0
votes
1answer
70 views

Inner product of function of period $2\pi$ with exponential

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous with period $2\pi$. Prove that $$\lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{j=1}^Nf\left(\dfrac{2\pi j}{N}\right)e^{-2\pi ...
2
votes
1answer
79 views

Fourier transform supported on compact set

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\hat{f}(y)=1_{[-\pi,\pi]}(y)\sum_{n=-\infty}^\infty f(n)e^{-iny}$$ in the sense of $L^2(\mathbb{R})$-norm ...
2
votes
1answer
114 views

Fourier series of $f(x)=1$ and $f(x)=x$

Would someone be kind enough to explain to me what would be the Fourier series of $f(x)=1$ and $f(x)=x$ on $[0,1]$? See, all the equations I can find are for intervals of the type $[-L,L]$. Now I ...
3
votes
1answer
38 views

Fourier coefficients converging

I'm thinking about this question, which has no answer yet despite being on a bounty and having 100+ views. Maybe it would be easier to start by asking this: Let $g\in C_0^{\infty}(\mathbb{R})$ ...
7
votes
1answer
223 views

Computing Fourier transform for $L^2$ function

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
3
votes
0answers
56 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
1
vote
0answers
78 views

Q: Bases and Frames using Fourier Series

Define $w: \Bbb R \rightarrow \Bbb C$ by \begin{equation} w(t) =\begin{cases} 1/\sqrt{2\pi} & t \in [0, 2\pi)\\ 0 & \text{otherwise}. \end{cases} \end{equation} and for $n \in \Bbb ...
3
votes
2answers
107 views

Evenness of Fourier coefficients implies even function

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$Assume that the Fourier coefficients determine an $L^1$ ...
2
votes
1answer
48 views

Evenness of Fourier coefficients

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is even if and only if ...
2
votes
2answers
94 views

Can I calculate approximately a definite integral of a function by integrating its Fourier Sine Series term-by-term?

I'm not sure how to put fancy formulae here because I'm a fairly new user. So bear with me for a moment as we go through a formulae-less reasoning. 1) I have a function $f(x)$. 2) I want to ...
1
vote
1answer
81 views

These questions are all about Fourier analysis.

Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: ...
12
votes
1answer
318 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
3
votes
1answer
91 views

A PDE problem about $(\partial_x^2 + \partial_t^2)u = 0$ using Fourier series.

I'm trying to solve the following initial value problem (from Folland, pg. 277, exercise $48$a) using Fourier series: Let $x \mapsto f(x)$ and $x \mapsto u(x, t)$ be periodic functions on ...
2
votes
0answers
175 views

Prove: $\int_0^{\infty}\left(\frac{\sin x}{x}\right)^2dx=\pi/2$

I am dealing exercise 12 in Chapter 8 of Rudin's Principles of Mathematical Analysis. Given the function $f$: $$f(x) = \begin{cases} 1, & \text{if $|x|\le\delta$} \\ 0, & \text{if ...
1
vote
0answers
45 views

Showing that $\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} \mathrm{G}_t(x-2\pi n)\in\mathbb{C}^\infty((0,\infty)\times\mathbb{R})$

Welcome everybody :) I need your help in answering the following question: Let $t > 0$ and $\mathrm{G}_t(x) = (2\pi t)^{-1/2}e^{-x^2/2t}$ Show that $$\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} ...
5
votes
2answers
116 views

State-of-art of the Discrete Fourier Transform

I would like to know what is the state-of-art in the research of the discrete Fourier transform. I have listed some questions to help answering, please add your own to make the list more ...
3
votes
4answers
233 views

Is a Fourier Series a continuous function?

My question relates to the properties of the Fourier series of a function, $f: \mathbb{R} \to \mathbb{R}$. I know from an elementary course in differential equations (for engineers) that, for all ...
1
vote
0answers
36 views

Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.

Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
1
vote
1answer
145 views

Parseval's identity

How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
0
votes
1answer
215 views

Exponential Form of Fourier Series

Problem Suppose $f$ is a continuous function on interval $[-\pi,\pi]$ such that $\sum_{n\in\mathbb{Z}} |c_n| < \infty$ where $c_n = \dfrac {1}{2\pi} \int_{-\pi}^\pi f(x)\cdot \exp(-inx)~dx$, the ...