1
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2answers
25 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
1
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0answers
9 views

When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
1
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0answers
15 views

Fejer's Theorem in relation to the Fourier Transform

I have this question that relates the Fejer theorem with the Fourier Transform. Any help would be appreciated. If $f$ is of moderate decrease then $$\int_{-R}^{R}\left(1-\frac{|\xi|}{R}\right) ...
0
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0answers
15 views

WHY?The Squared Euclidean Imbalance are equal to Fourier coefficients

I'm reading the classical paper about distinguishing attack, How Far Can We Go Beyond Linear Cryptanalysis ,Thomas Baign`eres, Pascal Junod, and Serge Vaudenay. The only proposition I don't ...
0
votes
0answers
13 views

Obtain the complex Fourier Series of the following function:

$$f(t)=t^3 \;\;\;\;\;\;\;\;\;\;\;\; -3/2<t\leq 3/2 $$ $$f(t)=f(t+3)$$ I've tried setting up an integral for $C_n$ coefficients using the formula $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ...
0
votes
0answers
12 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
3
votes
1answer
30 views

Evaluate the following Dirac delta integrals:

a) $ \int^{+\infty}_{-\infty} \delta'(t-\pi)e^{-t^2} \; dt$ b) $ \int^{+\infty}_{-\infty} \delta(-3t)(\frac{e^{-t^2}}{\ln(t^2 + 3)}) \; dt $ c) $ \int^{+\infty}_{-\infty} \delta(4t)\sinh{t^2} ...
1
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0answers
15 views

Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
0
votes
2answers
33 views

Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
2
votes
1answer
20 views

Fourier series with a weighted mean square norm

I am interested in Fourier series with a non-uniformly weighted error norm. What I mean by this is that the usual Fourier series of a periodic function is a minimizer of the mean squared error: $$ J_N ...
1
vote
1answer
21 views

Please help with this Discrete fourier transform question

Consider the ODE $\frac {d^2u}{dx^2} + 2\pi\frac {du}{dx} + \frac 54\pi^2u = g(x)$ where g is a periodic fuction with period 1 given by $g(x) = e^{\pi x}$ , $ 0 \le x \lt 1$. It is desired to find ...
0
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0answers
25 views

Triangular signal Fourier transform. Please help me [closed]

Please help me. I don't know how to solve part b) (the triangular signal fourier transform).
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votes
0answers
8 views

Nyquist Sampling Theorem

I'm working on the proof of the Nyquist sampling theorem. Mainly I'm wondering about the regularity conditions. In particular, suppose $f$ is continuous on $\mathbb{R}$ and $\hat{f}(k)=0$ unless ...
0
votes
0answers
10 views

Express the fourier coefficients of a autocorrelation function (complex form)

Here are some hints from the instructor: "Just plug the expression for the autocorrelation function into the formula for the Fourier coefficients, you get a double integral, and then smuggle in an ...
3
votes
1answer
39 views

Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases} $$ A problem in Stein's Fourier Analysis asks ...
0
votes
0answers
12 views

Nyquist Frequency, filter, phase/amplitude

Problem The problem seems quite simple and I believe it is though I have no idea how to approach it. I have tried googling 'Nyquist frequency' but have not had any luck with problems similar to this. ...
2
votes
1answer
50 views

Fourier Series/Parseval's Theorem

I have pretty much completed this question and have found the Fourier representation to be; $$ f(x) =\frac A2 +\sum_{n=0}^\infty 2A\frac{\cos(((2n-1)(\pi x))/2f_o)}{\pi(2n-1)} $$ Now I don't ...
1
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0answers
32 views

Uniqueness of Fourier Coefficients

I'm reading through Stein & Shakarchi's book on Fourier Analysis on my own, and have a question about the proof of the following theorem: Suppose that $f$ is an integrable function on the circle ...
2
votes
2answers
32 views

How does one derive the complex form of the Fourier series?

Specifically, I have gone from the Fourier Series in this form: $$\sum\limits_{n=1}^{\infty} a_n\cos(nx) +b_n\sin(nx)$$ and I have taken it to this form: $$\sum\limits_{n=1}^{\infty} \frac{(ib_n - ...
0
votes
2answers
44 views

$f$ is real valued iff $\overline{ \hat{f}(n) } = \hat{f}(-n)$

The problem I am considering is: For $f$ a $2\pi$-periodic and Riemann integrable function, show that $f$ is real valued iff $\overline{ \hat{f}(n) } = \hat{f}(-n)$. Here $\hat{f}(n)$ represents the ...
1
vote
1answer
26 views

Is $f(t-N t_0)=\sin(2\pi f_0t)\cos(2\pi f_1t)$ always true?

Is it true that multiplying two sinusoidal functions, always result in some periodic waveform. i-e $$f(t-N t_0)=\sin(2\pi f_0t) \cos(2\pi f_1t)$$ If so, then how can we calculate the period ( ...
1
vote
0answers
32 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 $[0,1]$. ...
1
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0answers
17 views

Decay of Fourier coefficients of $\frac{1}{f}$

Let $\alpha > 0$ and define \begin{equation*} \mathbb{H}^{\alpha}\left[-\pi,\pi\right] = \left\{f:\left[-\pi,\pi\right]\mapsto\mathbb{R} \;s.t.\; \sum\limits_{n\in\mathbb{Z}} \left\lvert ...
0
votes
0answers
22 views

Discrete Fourier Transform of the infinite series

I am reading this book and having hard time understanding how to get to eq(2) from eq(1) $$P(k,t) = e^{-\alpha t} \sum\limits_{l,m=-\infty}^\infty (-i)^m e^{ik(l+m)} I_l(\alpha t) I_m(i\beta t) ...
1
vote
2answers
32 views

About Fourier coefficient definitions

I'm studying Fourier analysis and my book gives the following definitions for the Fourier series and Fourier coefficients: Fourier series of $2\pi$-periodic function $f(\theta)$ is defined as: ...
1
vote
0answers
28 views

Uniform Boundedness in N of $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$

Show that $\int_0^\infty \frac{\sin(x)}{x}\,\mathrm{d}x = \frac{\pi}{2}$, and using that show that $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$ is uniformly bounded in N and ...
2
votes
1answer
34 views

Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
0
votes
0answers
25 views

Rate of convergence of Fourier series

I am having a bit of a confusion regarding convergence results. Suppose $f$ is Lipschitz, or $f \in C^\infty$ and let $S_{N}f$ be its truncated Fourier series. In the wikipedia page ...
1
vote
0answers
21 views

Fourier Transform of $sin(5t - \frac{\pi}{4})U(t+8)$

I have this function $$ sin(5t - \frac{\pi}{4})U(t+8) $$ I know the Fourier Transform of $sin(5t - \frac{\pi}{4})$, which is $$ \frac{e^{-\frac{\pi^2}{2}fj}}{2j}\left [\delta (f-\frac{5}{2 \pi}) ...
1
vote
0answers
23 views

Rearrangement of Fourier Series Sum to attain convergence

Let $f$ be a continuous function with diverging partial Fourier sums $S_N(f)(0)$ : $$ f(\theta) = \sum \limits_{k=1}^\infty \alpha_k P_{N_k}(\theta)$$ Let $f(x) \sim \sum \limits_{n=-\infty}^\infty ...
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 ...
0
votes
1answer
78 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 $H_{n}(x)$ ...
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 ...
2
votes
1answer
57 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
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
29 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
28 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
74 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
53 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
39 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 ...
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 ...
1
vote
1answer
53 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 ...
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
1answer
88 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 ...