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

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898 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 ...
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1answer
271 views

How to find the Fourier series of a periodic function

Find the Fourier series of the function $f(t)=3t^2$, $-1\le t\le 1$. How do I solve this problem? What is the general formula and the way to solve this?
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2answers
78 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} ...
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0answers
77 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 ...
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0answers
105 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) ...
2
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1answer
43 views

Why do Fourier Series only work from $- π$ to $π$?

Take Euler's famous example: $$\dfrac{1}{2} x= \sin x-\dfrac{1}{2} \sin 2x+ \dfrac{1}{3} \sin 3x- \dfrac{1}{4} \sin 4x+\cdots $$ What is the reason this only works on $[-π,π]$?
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42 views

Proof that these are Fourier coefficients

I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$ In Murphy's book ...
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0answers
35 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
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1answer
102 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 $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
2
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2answers
314 views

Why is the zeroth coefficient in a Fourier series divided by 2?

I just learned that $a_0$ is basically the average of a function $f(x)$ on the interval $[-\pi, \pi]$, and that a Fourier series is given by $$ f(x) = a_0 + \sum_{n=1}^\infty (a_n \sin(nx) + ...
2
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1answer
42 views

How can I find this integral for a fourier series?

I have to calculate the following integral $$ b_n = \dfrac{1}{\pi} \int_{-\pi}^{\pi} \dfrac{1}{2}x \sin nx dx$$ The correct answer is apparently $$\dfrac{(-1)^{n-1}}{n}$$ But I have no idea how I ...
3
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1answer
82 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} ...
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0answers
34 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 ...
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0answers
54 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
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2answers
1k 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
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1answer
83 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 ...
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0answers
37 views

Convergence of Fourier coefficients of a periodic function

Given a function $g: [0,\pi] \to \mathbb{R}$, if for example $g(0) = g(\pi) = 0$ and we write the odd and periodic extension of $g$ as a Fourier series $$ g(x) = \sum_{m=1}^{\infty} {\hat{g}_m \sin{m ...
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1answer
43 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 ...
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1answer
41 views

Is the Fourier Series correct?

Could you tell me if the following Fourier series of the function $f(x)=x^2, -\frac{L}{2} \leq x \leq \frac{L}{2}$ is correct?? $$$$ $$a_0=\frac{2}{L} ...
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0answers
54 views

Show sum involving sines is non-negative

I want to show that \begin{equation} \sum_{\substack{k \geq 1 \\ k \text{ odd}}} k e^{-k^2 a} \sin(kx) \geq 0 \qquad \text{for all } x \in [0,\pi], \, a > 0. \end{equation} How should I start? I ...
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1answer
45 views

Expanding unity in terms of orthogonal functions cos( alpha(i) * y)

It is written in the book I am reading without proof that if we expand unity in terms of orthogonal functions $cos( \alpha_i y)$, we get: (Please check this link) ...
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1answer
2k views

Proof of Parseval's Theorem for Fourier Series

Ok so I want to prove the above expression, I substituted the complex fourier series for f and using the fact f may be complex-valued, carried on by representing $|f(x)|^2$ as $f(x)f(x)^\ast$ where ...
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1answer
81 views

Termwise Integration of Fourier Series

This is a question from Edwards and Penney 4th edition Differential Equations and Boundary value problems from section 9.3. Suppose that $f(t)$ is a piecewise continuous period $2L$ funtion. ...
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1answer
234 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 ...
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0answers
42 views

This $\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$ integral: does a closed form exist?

$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$$ Does a closed form for the above exist, ideally for $n,m\in\mathbb{C}$ (most bounds probably removed at one point using ...
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2answers
41 views

Prove $\frac1T \int_0^T\left(\sum_{k=-\infty}^{\infty}c_ke^{j{\frac{2\pi kt}{T}}}\right)^2dt= \sum_{k=-\infty}^{\infty}|c_k|^2$

This question relate to fourier series in electrical engineering but I post it here as it's only mathematical concern. I cannot prove this $$\frac1T ...
2
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1answer
33 views

Contradiction between $a_0$ and $a_k$ for Fourier Series

I need to calculate the Fourier Series for the function $f(x) = |x| \; f:[-\pi,\pi] \to \mathbb{R}$ When calculating $a_k = {1 \over \pi} \int_{-\pi}^{\pi} f(x) \cos{(kx)} dx \; (k \in \mathbb{N_0})$ ...
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1answer
181 views

Hölder Condition for Fourier Series

So I'm trying to prove that the function (as represented by a Fourier series) $ f(x) = \sum_{k=0}^\infty 2^{-k\alpha}e^{i2^kx}$ satisfies the Hölder Condition: $|f(x+h)-f(x)| \le C|h|^\alpha$, with $0 ...
2
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1answer
148 views

Fourier decomposition of the Mandelbrot set

It is not clear that the boundary of the Mandelbrot set is an analytic curve, even though it is connected. Nevertheless, we can approximate the boundary with a curve by iterating a finite number of ...
2
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1answer
173 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 ...
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1answer
67 views

Calculating Fourier expansion using Legendre Polynomials

I'm trying to write any function of the type $t^m$ using Legendre polynomials $P_n(t)$ . That means: $$t^m=\sum_{n=0}^\infty\langle P_n,t_m\rangle P_n =\sum_{n=0}^\infty a_{mn}P_n$$ Where I have to ...
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0answers
51 views

Fourier expansion of the complexified Gram series

Consider the Riemann's R function, also known as the Gram series: $$\text{R}(x)=1+\sum_{k=1}^{\infty}\frac{\left(\log x\right)^{k}}{kk!\zeta(k+1)}$$ Now consider the form: $$\text{R}\left(e^{2\pi ix} ...
2
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1answer
64 views

Discontinuous functions with finite Fourier series approximation?

Yesterday I posted a question regarding the computation of complex Fourier coefficients for the functions $$f(t) = \sin(2 \pi t)$$ $$f(t) = |\sin(2 \pi t)|$$ where $0 \leq t \leq 1$. The first ...
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2answers
59 views

Finding complex Fourier coefficients

This is probably an easy question, but I'm a little bit stuck, so any help will be appreciated. PROBLEM Find the complex Fourier coefficients of: $$f(t) = \sin(2\pi t)$$ and $$f(t) = |\sin(2\pi ...
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2answers
123 views

Summing a series (from a physics problem)

How might we show that $$\sum_{k = 0}^{\infty}\frac{2}{2k + 1}e^{-(2k + 1)\pi x/a}\sin\left( \frac{(2k + 1)\pi y}{a}\right) = \tan^{-1}\left( \frac{\sin(\pi y/a)}{\sinh(\pi x/a)} \right) $$ where $x, ...
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0answers
49 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 ...
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0answers
45 views

Finding the number of derivatives for series problems

I have the following problem: How smooth are the following functions? That is, how many derivatives can you guarantee them to have? $$a)\;\;\;\;\; ...
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1answer
169 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 ...
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1answer
144 views

Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
4
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1answer
375 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
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2answers
149 views

Trigonometric series problem

I have the following problem from my Fourier analysis book I would need some guidance with. I have tried it, but apparently I made some mistakes...here is my problem: We have: $$\sin \theta ...
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2answers
92 views

Fourier Series - Integration

Could someone explain where I am going wrong with the following fourier series calculation please? I'm trying to compute the $A_{0}$ and $A_{n}$ coefficients for the fourier series: \begin{align} ...
2
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1answer
34 views

A relation for Fourier series

For $f$ and $f'$ in $L^2(0,1)$, define $e_k(x)=e^{2\pi ikx}$, $k \in \mathbb{Z}$. And define the Fourier series: $f=\sum _{k \in \mathbb{Z}}c_ke_k$, where $c_k=\left \langle f,e_k \right ...
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1answer
51 views

Creating a function which satisfies a given set of points

I have been tasked to write a program in matlab which will approximate a function $f(t)$ as a sum of sines and cosines given that it is defined in the domain $0 - 2\pi$. I have a set of points that ...
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2answers
52 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 - ...
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1answer
62 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
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2answers
67 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 ...
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1answer
26 views

Fourier Coefficients of a Sequence of Functions

Let $f_k$ be a sequence of Riemann integrable functions over $[0,2\pi]$ such that $$\lim_{k\rightarrow\infty}\int_0^{2\pi}|f_k-f|=0$$ for some function $f$. Let $\hat{g}(n)$ denote the $n$th Fourier ...
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0answers
68 views

How to find the value of this sum?

The sum below numerically (to 13th digit at least) is the same as $\ln 2$. So there should be a way to prove it analytically, but I haven't succeeded. Any suggestions? ...
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1answer
41 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 ( ...