Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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7
votes
2answers
186 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
2
votes
1answer
80 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 ...
2
votes
0answers
21 views

Inequality on $L_1$ norms of tirgonometric polynomials generated with a smooth function

Let $\varphi\in C_0^\infty(\mathbb R)$ and for $n\ge1$ $$ f_n(x)=\sum_{k=-\infty}^\infty \varphi(k/n)e^{i k x}. $$ I seem to remember that there is an inequality $\|f_n\|_{L_1(\mathbb T)}\le C$, where ...
0
votes
1answer
79 views

How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
3
votes
1answer
84 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 ...
0
votes
0answers
14 views

Relating Fourier Transform to an Integral involving Sin(vt)

I have data for a function $S(Q)$ and I'm trying to find values for a different function $g(r)$ Now I know $g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ$ This is closely related to the sine ...
1
vote
1answer
29 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
0answers
43 views

How to solve this differential equation of the second order

Do you know how to solve this equation? I'm a physicist student and I have initial equation, condition and answer. Unfortunately I need an explanation how this answer was got. I am mew to such ...
2
votes
1answer
143 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
2
votes
1answer
58 views

For a given sequence $(a_k)$, there is no Riemann integrable function f such that $\hat{f}(k) = a_k \forall k$

I'm working out of Stein's Fourier Analysis: An Introduction, and am on chapter 3. There is an exercise that gives us a specific sequence $(a_k)$ and asks us to show that ...
0
votes
0answers
32 views

fourier series and correlation coefficients question?

We have the signal in the figure. I must do the trigonometric fourier series of the signal and also the exponential fourier series.Also,find the correlation coefficients between $f(t)$ and $e^{3t}$. ...
0
votes
1answer
30 views

please help me understand the lecture note? heat equation and fourier series

I don't quite understand equation 3.73 and 3.74. To get $T(x,t)$ I thought I had to multiply F and G. How does that give equation 3.73? I got G as e^{stuff} as in the last bit of equation 3.73. ...
0
votes
2answers
31 views

Heat equation. Find $B_n$ (using boundary conditions?)

Can anyone help me with (b)(i)? I've done the first part of it. I've tried putting some boundary conditions in but cannot find $B_n$ What fact should I use? An infinite slab of material, of ...
0
votes
1answer
40 views

Fourier transform real and imaginary part question?

I have to find the fourier transform of $f(t)=e^{-a^*t}*u(t)$ For a>0 the signal has an infinite value therefore doesnt have a Fourier transform.For a>0 we have: ...
0
votes
1answer
30 views

Fourier series/transform with cutoff

(1) The span of the trigonometric polynomials is $span \{ e^{i n x}\ |\ n \in \mathbb{Z} \} = L^2([-\pi,\pi])$. Is there any nice way of characterizing the span with a cutoff, namely: $span\{ e^{i n ...
1
vote
0answers
30 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
0
votes
1answer
30 views

2-D Fourier Transform of complex exponential with 2-D quadratic phase

I've been looking around to see if there is either an exact transform pair or an approximation to either of the following but have not been able to find anything: $$ \mathcal{F}_{xy}\left( ...
2
votes
1answer
58 views

Big O proof of Fourier Coefficient

Let $f(x)$ be a $2\pi$ periodic function on R. Assume that Hölder continuous: $$\sup_{x\neq y} \frac{|f(x)-f(y)|}{|x-y|^{-\alpha}} \leq C$$ for some constants $C$ and $\alpha \in \,]0,1]$. Prove ...
3
votes
1answer
75 views

Why should we use the Fourier Transform?

I'm a CS/Math double major, and during my study (and reading sources out of my own interest) I've had some encounters with the Fourier Transform. I understand the theory behind Fourier series, and ...
1
vote
1answer
54 views

Fourier Series of a decaying Cosine

I'm trying to find a handy way to find the infinite sum $ \sum_{n=1}^\infty \frac{cos( a n)}{n^2} $ through a Fourier series. The regular sum can be evaluated using Mathematica fairly easily, and ...
0
votes
1answer
41 views

Solving the heat equation using Fourier series; specific questions

Like this previous question, Solving the heat equation using Fourier series, I too am reading the same wikipedia article, ...
5
votes
1answer
58 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
0
votes
0answers
12 views

2D Wave propagating in duct with height change

Suppose that we have a two - dimensional rigid wall duct cosisting of two semi - infinite regions $x<0,\ 0\leq y\leq a\ $ and $x>0,\ a<y\leq b$ (this means exactly that there is a height ...
2
votes
0answers
30 views

Phase speed of 2D wave

I'm a little stuck with understanding the properties of 2D waves. I have the wave $e^{2\pi i(jx+ky-\omega_{j,k}t)}$=$\exp\left(2\pi i\left(\left[\begin{array}{l} j \\ k ...
5
votes
1answer
60 views

An $L^1$ function whose Fourier series converges but not to itself

Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
0
votes
1answer
26 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?
0
votes
0answers
24 views

Decay rate of fourier coefficients obtained from discrete fourier transform.

We know that for a smooth function the fourier coefficients obtained from the continuous fourier transform decays at an exponential rate. Is the same true for fourier coefficients obtained from the ...
0
votes
2answers
99 views

What is the advantage of using Fourier Series representation rather than the function itself?

Suppose we have a function $f$ defined over $[a,b]$ to the real numbers, i.e. $f: [a, b] \to \mathbb R.$. We can approximate this function as Fourier Series. Suppose $a_n, b_n$ is the Fourier series ...
7
votes
4answers
162 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 $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
0
votes
0answers
18 views

Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$

This is a question from a book that I'm trying to solve and I don't know how. Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$ Can you please give me some ...
1
vote
0answers
51 views

Proving periodicity of sine and cosine [duplicate]

If we define the sine and cosine functions by their Maclaurin expansions, how do we prove they are periodic with period $2\pi$?
0
votes
1answer
27 views

Given Fourier series for $f(x)$ continuous over $[-\pi, \pi]$. find $\int_{-\pi}^\pi f(x)\cos^2(nx)dx$

I'm learning to the exam and I find this exercise in the book , and I can't think how I solve it. Given Fourier series for $f(x)$ continuous over $[-\pi, \pi]$. $$f(x) \approx \frac{a_0}{2} ...
5
votes
1answer
124 views

Jacobian of Fourier Transformation

I am trying to calculate the Jacobian determinate of the Fourier transform which I stumbled upon when studying the Path Integral in Quantum Field Theory. I know the answer should be $1$ but I don't ...
1
vote
2answers
58 views

Find complex Fourier coefficients

let $f(x) = \sum^{10}_{m=1}(-1)^m \sin(2^m x)$. denote complex Fourier coefficients of $f(x)$ over $[-\pi, \pi]$ as $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
1
vote
0answers
29 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}}, ...
0
votes
0answers
18 views

Determine whether the set is uniqueness set

We say that $\Lambda$ is a uniqueness set for the Paley-Wiener space $PW_{\pi}$ if $$(F \in PW_{\pi} \wedge F|_{\Lambda}\equiv 0) \rightarrow F\equiv 0.$$ For example, $\Lambda =\mathbb Z$ is a ...
2
votes
2answers
49 views

Dirac Comb Times Step Function

Can someone explain me what is the effect of the Heaviside step function $\Theta(t)$ on a Dirac Comb (Fourier series)? $$ \left[\,\,\sum_{n=-\infty}^{\infty}c_{n}\,\delta\left(t - nT_{0}\right) ...
1
vote
1answer
55 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
78 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 ...
0
votes
1answer
34 views

$f(x) =\cos(x-y) -\cos(\delta)$ plotting

Ok, so this is a confusing one. I'm not sure what my teacher is looking for. The problem is: Plot any number $-\pi < y< \pi$ and pick a small number $\delta > 0$ so that the whole interval ...
1
vote
4answers
122 views

What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$?

I'm pretty sure that there's a theorem that says that the Fourier coefficients of a sum of $\cos(nx)$ and $\sin(nx)$ 's are the coefficients of the sum itself. I tried to prove that in the specific ...
1
vote
1answer
91 views

Convergence of the Fourier serie of $f(x)=e^{2\pi i \alpha x}$

I have some difficulties with the last part of an old exam exercise. For the 1-periodic function $f$ defined on $[0, 1[$ by $f(x)=e^{2 \pi i \alpha x}$ with $0<\alpha <1$. I have found that its ...
2
votes
1answer
49 views

Finding Fourier coefficients?

The question is as follows: $f(x) = \cos(\pi x)$, $g(x) = f(x+2010)$. I need to find the sum of all of $g$'s Fourier coefficients from $-\infty$ to $\infty$. I know that $f=g$. Therefore $g$'s $n$th ...
4
votes
2answers
271 views

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series)

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series). I have , somehow, to find the sum of $\sum_{n=1}^\infty \frac{1}{n^4}$ using Parseval's theorem. I tried ...
1
vote
1answer
25 views

Proving uniform convergence with some kernel

Question Given $K_n=\cases 0$ elsewhere , $n- n^2|x|$ for $x<|\frac 1n|$ , $f$ is continuous, $2\pi$ periodic $\Bbb R \to \Bbb C$ . $f_n(x)=\int _{-\pi}^ \pi f(t)Kn(x-t)$ prove that ...
0
votes
1answer
44 views

Fourier series verification

Question: $$f(x)= \sum_{n=0}^\infty \frac {e^{inx}}{1+n^2}$$ if $x\ne 2\pi k$ and $f(x)=0$ if $x=0 , x=2\pi k$ Find $\hat f(n)$ Find the Fourier series of $\displaystyle g(x)=\int _0^xf(t)dt$ ...
6
votes
3answers
174 views

Why does the Fourier series of $x$ not seem to give the right value?

I'm reading a lecture about Fourier series , and it says that you can represent any continuous function as Fourier series. There's a given example: Let $f(x) = x$. $f(x) \approx ...
0
votes
1answer
36 views

Discrete fourier transform problem

We have taken $1000$ observations from signal $s(t),t \in\Bbb{R}$ $$h(k)=s(k\Delta t+t_0),k=0,1,\dots,999,$$ where $\Delta t=1/200 $ and $t_0=-2$ (in seconds). When we calculate discrete fourier ...
0
votes
1answer
35 views

Convergence of a fourier series of $f(x)=1+\sin \frac {\pi^2}x$

Question: let $f:\Bbb R \to \Bbb R, f(0)=1 \forall x\in[-\pi,\pi] \setminus \lbrace0\rbrace , f(x)=1+\sin \frac {\pi^2}x$ Does the fourier series of this function converge at zero? If it does what is ...
2
votes
2answers
59 views

Fourier Series.

I have found the Fourier Series of $$f(x)= \begin{cases} 0\colon & -\pi<x<0 \\ x\colon & 0<x<\pi \end{cases}$$ to be equal to: $$f(x)=\frac{\pi}{4}+\sum_{n=1}^\infty ...