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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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} ...
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1answer
25 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
40 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
33 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
52 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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0answers
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How can we derive $\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$?

When I was reading digital communication theory, I couldn't derive following equation $$\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)] $$ ...
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2answers
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$f$ 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
23 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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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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Expanding a piecewise defined function, what will the series converge to at $x=-1,0,1$? [closed]

If we expand $$f(x)=\begin{cases} (x+1) & -1<x<0; \\ -x & 0<x<1 \end{cases}$$ what will the series converge to at $x=-1$, $x=0$, and at $x=1$? Hey I tried to work this out on ...
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1answer
33 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 ( ...
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1answer
86 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
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1answer
36 views

Is it possible to solve a system of equations comprising FFTs?

Consider the following known matrices, A, B, C and these unknown matrices X,Y, all of which comprise values in the Real domain. Also consider $F(x)$ as the *Fast Fourier Transform function* (the ...
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55 views

How did Fourier find the formula for the fourier series coefficients?

The modern proof use the dot product but did he use that ?
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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 $[0,1]$. ...
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PSD Rational function to power series

I have a Power Spectral Density given as a Rational function that is: \begin{equation} \phi(e^{i\omega}) = 1/(1-1.7464e^{-i\omega}+1.2602e^{-2i\omega}-0.4366e^{-3i\omega}+0.625e^{-4i\omega})^2 ...
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0answers
20 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 ...
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Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
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1answer
20 views

Fourier coefficients.

I don't quite see how the following hold and would appreciate an explanation: (1) The Fourier coefficients of $cos(\frac{6\pi n}{N})$ are $\delta[k-3]+\delta[k+3]$ (2) ...
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2answers
98 views

Proof of Wirtinger inequality

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ ...
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4answers
148 views

Find the Exact sum

Give the fourier series representation of $f(x) = x$ on $[-\pi, \pi]$. Use the result to give the exact sum of... $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}$$ $$\text{ where } x \in [-\pi,\pi]$$
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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) ...
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Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
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2answers
44 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: ...
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2answers
54 views

Find the fourier series for $\cos^{2N}(\theta )$.

I'm working my way through a book for prelim prep and found the problem: Find the fourier series for $\cos^{2N}(\theta )$. The hint is to not use integrals but the only method I know involves ...
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2answers
124 views

If square waves are square integrable, why doesn't fourier expanding work?

If square waves are square integrable, then why does expanding on a fourier basis not recover the equation?
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1answer
40 views

Real fourier series of $e^x$ on $(-l, l)$

The complex Fourier series is: $$\sum_{n=-\infty}^{\infty}(-1)^n \frac{l+in\pi}{l^2+n^2\pi^2}\sinh(l)e^{in\pi x/l}$$ How can I derive the real Fourier series (sines and cosines) from this? Do I just ...
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1answer
98 views

Fourier series for $e^x$

I'm trying to teach myself partial differential equations from Strauss' book. I have run into a very bizarre problem - I cannot figure out what is the Fourier series of $e^x$! And not even Google has ...
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0answers
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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 ...
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1answer
28 views

Properties of periodic functions

Let $f$ and $g$ be periodic functions of period $p$. Then $af(x)+bf(x)$ with $a,b$ constants and $f(x)g(x)$ are both of period $p$ I'm not exactly sure how to prove these properties of periodic ...
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1answer
41 views

Why do the first spikes in these plots point in opposite directions?

With the following Mathematica program: ...
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1answer
43 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 ...
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3answers
45 views

Fourier Series with Complex Exponentials

In my Signals and Systems class, we learned that the Fourier Series of a signal $x(t)$ is given by $$ x(t) = \sum_{k = -\infty}^{\infty} X_k e^{ik\omega_0t} $$ where $\omega_0 = 2\pi/p$ and $$ X_k ...
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1answer
36 views

Expansions onto “bases”…?

When we consider expanding functions into fourier series, or taylor series, or onto the spherical harmonics-are these projections onto a basis? Are these bases complete? How can we show this? I know ...
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40 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 ...
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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}) ...
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29 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 ...
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0answers
66 views

Is this wave noisy at prime powers and silent at composite numbers?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ And the von Mangoldt function should then be: $$\Lambda(n)=\lim\limits_{s ...
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1answer
79 views

Use Fourier's method of separation of variables to solve the boundary value problem

Use Fourier's method of separation of variables to solve the boundary value problem comprising the following PDE and BC: PDE: $x \sin(y) u_x + \cos(y) u_y = -2 \sin(y) u $, $u = u(x,y)$ Boundary ...
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92 views

Pointwise convergence of Fourier sine series and uniform convergence of Fourier cosine series.

Let $\overline{f}$ be a function on the whole real line, such that $\overline{f}$ is continuous and differentiable everywhere, and its derivative $\overline{f}'$ is also continuous everywhere. Now, ...
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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 = ...
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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 ...
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0answers
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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 ...
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115 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 $H_{n}(x)$ ...
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1answer
112 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 ...
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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 ...
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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 ...
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44 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
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1answer
144 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
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1answer
64 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 ...