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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2
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1answer
71 views

What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
2
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0answers
22 views

Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
0
votes
4answers
182 views

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
4
votes
2answers
90 views

Fourier series for $\sec(x)$

Expand in Fourier series the function $$f(x)=\sec(x) \quad x\in(-\pi/4,\pi/4).$$ Hint: Deduce a relation between the coefficients $a_n$ and $a_{n-2}$ Since this function is even, $b_n=0$ and ...
3
votes
2answers
71 views

Replicating Kolmogorov's Counterexample for Fourier Series in Context of Fourier Transforms

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More ...
9
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2answers
146 views

$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
0
votes
1answer
7 views

Increasing order of fourier coefficients on the boolean cube

Given a function $f:\{0,1\}^n\rightarrow \{0,1\}$, is it true that for any $S,T\subseteq[n]$, such that $S\cap T =\phi$, then $\hat{f}(S\cup T)\leq \hat{f}(S)$? It seems so to me cause, if if you just ...
1
vote
1answer
41 views

Function Integrable in an improper sense that does not satisfy Riemann's Theorem

I need some help over the subject of Fourier series... Do you know if there's a function $g(t)$ integrable in a improper sense over an interval $[a,b]$ and such that $\lim\limits_{p\rightarrow ...
2
votes
1answer
47 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
2
votes
1answer
55 views

Solving wave equation by fourier method

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
7
votes
1answer
98 views

Sets of Divergence for Fourier Partial Integals

It is a consequence of Carleson's theorem together with a transference argument that (see Section 4.3.5 in L Grafakos, Classical Fourier Analysis for proof) that the Fourier partial integrals of a ...
0
votes
1answer
29 views

When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
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0answers
23 views

Fourier series for $x$ in $(0,\pi)$ for the complete family $\{ cos(kx) \}_{k \geq 0}$

So I have to obtain the fourier coefficients $C_k$ so that: $x= \sum_{k=0}^{\infty}C_k\cos(kx) $ $x \in{[0,\pi]}$ and for $k \in \mathbb{N_{0}} . $ I have used ...
0
votes
1answer
21 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
0
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0answers
15 views

Find a derivative of equation that contains Fourier series

I need to find a derivative of follow equation $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ I know the derivative of $\left(r_{0} + \sum ...
1
vote
1answer
38 views

An equality about Fourier transform

I have read an equality about Fourier transforms which I can not proof. It is as following: Let $u\in C_0(\mathbb{R}^n)$ and \begin{equation} g(x_1,x_2,...,x_{n-1}):=u(x_1,x_2,...,x_{n-1},0). ...
1
vote
1answer
51 views

What if the Fourier series of a periodic function also has periodic coefficients $a_k$

If given that $x(t)$ is a periodic continuous time signal, with periodic $T$. It can be expressed by the Fourier series, i.e. $x(t)=\sum\limits_{k=-\infty}^{+\infty}\,a_k\cdot e^{j k \frac{2 ...
2
votes
0answers
24 views

Fourier Transform by hand

For an exam we have to calculate Fourier Transform by hand in complex space and in $\mathbb{Z}_{32}$ space ($\mathbb{Z}$ mod 32). I am familiar with recursive algorithm in a complex space (example in ...
1
vote
1answer
40 views

Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
0
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0answers
43 views

How to motivate those expansions?

I've been reading a paper where the author needs to solve the biharmonic equation on the plane. In truth, the function being saught is a function $v$ such that $v = \nabla \times U$ and $\nabla^4 U = ...
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0answers
28 views

A sufficient condition for a series of functions to be $\mathcal{C}^i$ (Differentiable)?

Suppose we have the Fourier Series : f(x)=$\sum_{k=1}^{\infty} C_k f_k(x)$=$\sum_{k=1}^{\infty} C_k \sin(kx)$ defined in $(a,b) \in \mathbb{R}$ Using Dirichlet criterion I have shown the sum is ...
1
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1answer
21 views

Bessel equation of half-order (asymptotic)

Not really optimistic about getting a reply for a question tagged under "Bessel function" but here goes, I have $$J_{\frac{1}{2}} = (a_1 \cos(z) + a_2 \sin(z))Z^{-\frac{1}{2}} $$ and ...
1
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1answer
67 views

What is $\lim_{n \to \infty} n^3 a_n$? [duplicate]

$a_n$ is the Fourier coefficient of $$f(x) = \left(1 - \frac{|x|}{\pi}\right)^4$$ The answer is infinity, but can someone give an answer that doesn't require explicit computation of the $a_n$? I'm ...
3
votes
1answer
48 views

A Hölder continuous function whose Fourier coefficients do not decay very fast

At Stein's book of Fourier analysis (Chapter 3, page 91, exercise 15) I was trying to solve the following problem I have to prove that the result ...
3
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0answers
43 views

Using a Fourier Series to Solve Differential Equation

The problem states to use the fourier series of the function f(t) defined as follows: $f(t)= t+1 , -1<t<0 $ $f(t)=1-t , 0<t<1$ to solve the differential equation: x''+4x=f(t), x(0)=1, ...
1
vote
1answer
83 views

“Counterexample” for a weaker version of Riemann–Lebesgue lemma

My teacher gave us this version of Riemann–Lebesgue lemma in class: Let $g(t)$ be an absolutely integrable function on $[a,b]$, then $$\lim_{p\to\infty} \int_a^b g(t)\sin(pt)dt=0$$ Similarly for ...
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0answers
16 views

The bond between Fourier Transform and Epicycle theory

Can someone help me understanding the bond between the Fourier Transform and the epicycle theory? I have searched in many places such as: http://math.stackexchange.com/a/72479/185138 ...
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0answers
6 views

A theorem regarding epicycles

Can somebody help me understanding the theorem on the last page of that article about Fourier Series and Epicyles? ...
1
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1answer
64 views

Find the half range cosine fourier series expansion for $f(x)=(x-1)^2,\quad 0<x<1$.

Find the half range cosine fourier series expansion for $$f(x)=(x-1)^2,\quad 0<x<1$$ and hence deduce that $$\pi^2=8\left(\frac 1 {1^2}+\frac 1 {3^2}+\frac 1 {5^2}+\ldots\right)\tag{1}$$ My ...
1
vote
1answer
52 views

Finding the limit that involves Fourier coefficients,

Given the function $f(x) = 1 - \dfrac{|x|}{\pi}$, I had computed its Fourier coefficients, using integration by parts and got: $$ a_n = \begin{cases} 0, & \text{for $n$ even}, \\[6pt] ...
1
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1answer
34 views

Fourier series and transform (epicycles)

Let $\gamma:[a,b]\to\mathbb{C}$ be a continuous curve. 1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times ...
3
votes
2answers
99 views

Why is $\sigma_1(0)$ not $-\frac{1}{12}$?

The Eisenstein series $\mathbb{G}_2$ is given by $$\mathbb{G}_2(z) = -\frac{1}{24} + \sum_{n=1}^\infty \sigma_1(n) q^n$$ with $q=e^{2\pi i z}$ and $$\sigma_1(n):=\sum_{d\mid n} d$$ for $n\in\mathbb ...
0
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0answers
16 views

FFT differential equations

Given a generical differential equation what is the procedure to solve it using fft command. Can anyone explain me how to do it? For example: $$\frac{d^2y}{dt^2}+10\cdot \frac{d\:y}{dt}=-5\cdot ...
2
votes
1answer
46 views

Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for ...
1
vote
1answer
29 views

Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
0
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0answers
25 views

relation between $\zeta(2)$ and the fourier transform of $x^2$

I have problem with see the relation between the transform of $x^2$ in $[-\pi,\pi]$ and the function $\zeta$ de Riemann in the point 2, this say that using the transform fourier of $x^2$ prove that ...
0
votes
1answer
34 views

Applying Fourier transform to heat equation with source

I haven't had any experience with applying of FT to heat equation with source. But this popped up in an exercise. Any help in the right direction would be great. Consider: $$\frac{\partial ...
0
votes
1answer
40 views

Fourier transform of a piecewise

How should I go about seeking the Fourier transform for the piecewise function: $$f(x) = \left\{\begin{matrix} 0 ,&|x|>a \\ 1 ,&|x|<a \end{matrix}\right.$$ Is this the correct ...
1
vote
1answer
18 views

Fourier series for function

Consider the function f(x) = |x| $ - \pi \leq x < \pi $ Compute its Fourier series. $ a_{0} = \frac{1}{\pi} \int_{-\pi}^{\pi}|x| dx = \frac{2}{\pi} \int_{0}^{\pi} x dx $ I ...
0
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0answers
32 views

Signum function and Fourier transform

I'm extracting a portion of my notes which I believe I might have copied wrongly. Given this equation: $$\frac{G(\omega)}{2ic\omega} [e^{ic\omega t}-e^{-icwt}]$$ I want to find the Fouerir ...
1
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1answer
20 views

different formulas for the fourier series.

Quick question, I see both of these $$f(w) = a_0 + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw) \quad f(w) = \frac{a_0}{2} + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw)$$ Why the difference ( ...
1
vote
1answer
49 views

Parseval identity for $L^2(a,b)$?

The Parseval Identity states that: $\sum_{n=-\infty}^{\infty}|c_{n}|^2= \frac{1}{2 \pi} \int_{-\pi}^{\pi} |f(x)|^2 dx$ Where $\{c_{n} \}$ are the Fourier coefficients of f. Is there a general ...
1
vote
2answers
19 views

Fourier series function

$f(x) = x$ , $f(x+2\pi) = f(x) $ on $ [-\pi , \pi] $ How do I know that this function is even or odd? My book says odd, but I don't understand how to work this out? also why does $a_0 = 0$ ...
0
votes
0answers
24 views

Is $\sin(kx) $ a complete system in $L^{2}(0,b)$ for every k in $\mathbb{R}$?

Is the family $\{ \sin(kx ) \}$ a complete system in $L_{2}(0,1)$ for every $ k \in \mathbb{R}$, $k\geq1$ ? And less specifically, Is it a complete system in $L_{2}(0,b)$ for every b$\in \mathbb{R}$ ...
1
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2answers
66 views

Fourier series: Why is there a separate formula to determine $a_0$?

$$a_n = \frac 1 \pi \int_{-\pi}^{\pi} f(x) cos(nx) dx \quad\quad n\ge1 $$ Now I am wondering why there is a separate formula for $a_0$: $$a_0 = \frac 1 \pi \int_{-\pi}^{\pi} f(x) dx$$ It looks ...
0
votes
1answer
43 views

Fourier coefficients of $\cos(x/2)$

Is there a straightforward way to calculate the fourier coefficients of $\cos(x/2)$ in closed form on the interval $[0,2\pi]$? (I mean in terms of the generic $n$) From a calculation of the first ...
0
votes
1answer
32 views

Prove that the following function is $C^\infty$ in the point $\xi=0$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any way how to prove this? i think that i must use ...
0
votes
0answers
41 views

What do real and imaginary parts of phase spectrum represent?

In frequency domain, we can compute phase spectrum of a signal. Usually phase spectrum is complex valued. So my question is what do real and imaginary parts of phases of phase spectrum represent ? ...
0
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0answers
36 views

Approximation of the coefficients of the Fourier Series via the FFT

Is there literature on the approximation of the coefficients of the Fourier Series via the FFT? The approach I'm interested is merely numerical, consisting of computing the integrals with the ...
1
vote
0answers
27 views

Fourier Series Divergence

Define $b_n = \int_0^{\pi} \cos (nx).\sqrt{x} ~ dx $. Does the following series converge? $$ F(x) = \sum_{n=1}^{\infty} b_n \cos(nx) $$