4
votes
1answer
104 views
+50

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all ...
1
vote
1answer
40 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
0
votes
1answer
19 views

Rank of the harmonics in a Fourier series expansion

Let $\boldsymbol{A}(t)$ be a $T$-periodic matrix with rank $r$, and $\boldsymbol{A}_n$ the harmonics of its Fourier series expansion, so that $$ \boldsymbol{A}(t) = \sum_{n=-\infty}^{+\infty} ...
0
votes
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 ...
6
votes
3answers
189 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 ...
1
vote
1answer
26 views

transformation of DFT matrix

$\mathbf{F}$ is a unitary DFT matrix where the $(m,n)$-th entry of $\mathbf{F}$ is given by $\frac{1}{\sqrt{M}}e^{-\imath2\pi(m-1)(n-1)/M}$. Note that $\imath=\sqrt{-1}$. Let $\mathbf{A}$ be a matrix ...
3
votes
1answer
36 views

Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
1
vote
0answers
77 views

Q: Bases and Frames using Fourier Series

Define $w: \Bbb R \rightarrow \Bbb C$ by \begin{equation} w(t) =\begin{cases} 1/\sqrt{2\pi} & t \in [0, 2\pi)\\ 0 & \text{otherwise}. \end{cases} \end{equation} and for $n \in \Bbb ...
1
vote
0answers
21 views

Is it possible to calculate a single frequency bin in $O(\log N)$ time (considering the $N\log N$ performance of FFT algorithms)?

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...
0
votes
1answer
32 views

Does this transformation have an inverse?

Let $f(n)$ be a complex sequence. Then for prime $p$ define $\hat{f}(p) = \sum_{n = 1}^{\infty} a_n e^{-i 2 \pi n / p}$. Then let the transformation of sequences be $T$, i.e. $Tf = \hat{f}$. Is ...
2
votes
0answers
35 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
2
votes
2answers
289 views

Approximating $|x|$ by a linear combination of $1, \cos x, \sin x, \cos 2x, \sin 2x$

Let $\phi(x) = |x|$ for $x \in (-\pi, \pi)$. Suppose we approximate $\phi(x)$ by a linear combination of the functions $\{1, \cos x, \sin x, \cos 2x, \sin 2x\}$. What linear combination of the form: ...
2
votes
2answers
417 views

The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series

So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this. I know 'big ell 2' and 'little el 2' are ...
7
votes
4answers
325 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
0
votes
0answers
64 views

Dimension of space of band-limited, periodic, real functions

Dear all, I'm interested in the space of functions of $d$ variables which can be put in the following form $$f(x_1, \ldots, x_d) = ...