Tagged Questions
1
vote
0answers
62 views
Fourier Analysis of Prime Counting Function
I was thinking about the following:
Denote $\pi(x)$ as the prime counting function such that:
$$
\pi(x) = \#\text{ of prime numbers}\leq x
$$
It is well known from the prime number theorem that
$$
...
-1
votes
1answer
69 views
The maximum absolute value of DFT of window vector
Let x=[1, ⋯ ,1, 0, ⋯ ,0] be a window vector of length N, which consists of B consecutive 1s and the remaining N-B consecutive 0s.
I took the N-point DFT on x and got X=[X_0, X_1, ⋯, X_(N-1)] which is ...
0
votes
0answers
77 views
On Cooley–Tukey FFT algorithm
Does anyone know why Cooley-Tukey
FFT algorithm has a complexity $O(N\log N)$ for a sequence of length
$N$?
Thanks for any helpful answers.
1
vote
0answers
68 views
A question on algorithm complexity
It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$
for a signal with bandwidth $N$. How to see or show that the fast
Fourier transform ...
2
votes
1answer
88 views
The digit base and the NTT convolution
Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
6
votes
1answer
195 views
An elegant non-technical account on the work of Joseph Fourier.
It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
2
votes
0answers
170 views
What are the local minima in this spectrum?
Edit 6.2.2012: The sequence to be transformed should be f = 0,1,2,3,4,5... which makes the mentioning of the von Mangoldt function less necessary.
Edit 5.2.2012: I had the wrong plot of the insignal. ...
3
votes
0answers
242 views
How plot the Riemann zeta zero spectrum with the Fourier transform? [closed]
This question is closed as off-topic and duplicate of How plot the Riemann zeta zero spectrum with the Fourier transform in Mathematica?
In the paper "The Riemann Hypothesis" by J. Brian Conrey ...
2
votes
0answers
152 views
The discrete Fourier transform of a Dirichlet charachter
I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language.
I am trying to find the ...
15
votes
4answers
543 views
Interpretation of Poisson Summation Formula
This question arises from a Fourier transform class I took about a year back.
The poisson summation formula is:
$$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - ...
