## Fourier analysis – Determination of frequency components using FFT

The function defined below has three frequencies 2, 4, and 10. How can the Fourier transform (FFT) be used to display these frequencies?

``````   myfun(t_) =
1/(2 Sqrt(
2)) ((Sqrt)Abs(
3 + Cos(4 t) -
4 Cos(2 t) (-1 +
E^(-0.018` t) (Cos(9.99999594999918` t) +
0.0009000003645002213` Sin(9.99999594999918` t))^2) +
4 E^(-0.018` t) (Cos(9.99999594999918` t) +
0.0009000003645002213` Sin(9.99999594999918` t))^2))
``````

## nt.number theory – Fourier transform of \$ I_Y \$, \$ Y = { text {numbers with many prime factors} } \$

To let $$Y$$ Let be the set of integers $$N with more than $$D log log N$$ Prime factors. We can think, say, $$D = ( log log N) ^ {1- epsilon}$$,

We have pretty accurate approximations for the size of $$Y$$ (I am aware of Chapter II.6 in Tenenbaum's book and the references it contains.) I wonder what work is available there for the Fourier transform $$widehat {1_Y}$$ the characteristic function of $$Y$$,

I would expect $$widehat {1_Y}$$ To have tips on the main arches (ie bows around rations $$a / q$$ with a small denominator). That's because $$Y$$ is obviously "biased towards divisibility" and should therefore be slightly over-represented in the congruence class $$0$$ mod $$d$$ given for everyone $$d$$, relative to other congruence classes mod $$d$$,
A back-of-the-envelope calculation suggests that the value at the peak is around $$a / q$$ should be roughly proportional $$c ^ { omega (q)} / q$$, But what is known?

## fa.functional analysis – Characterization of a subset of the Sobolev space \$ H ^ k (0.2 pi) \$ in Fourier series

To let $$A: = {u in H ^ k (0.2 pi): u ^ {(j)} (0) = u ^ {(j)} (2 pi) mbox {for} j = 0,1, ldots, k-1 }$$, from where $$H ^ {k} (0.2 pi) subseteq L ^ 2 (0.2 pi)$$ is the Sobolev order room $$k$$ on $$(0, 2 pi)$$, Can we say that? $$u in A$$ iff
$$sum_ {n = – infty} ^ infty (1 + n ^ 2) ^ k | has {u} (n) | ^ 2 < infty?$$
In the above series $$has {u} (n)$$ are the Fourier coefficients of $$u$$, We think the answer to this question is positive because we may be able to identify $$A$$ with the Sobolev space of the Torus $$H ^ k ( mathbb {T})$$Use this result. But we can not show that there is an isomorphism $$A$$ and $$H ^ k ( mathbb {T})$$,

Know a reference for a characterization of $$A$$ with Fourier series?

## Fourier analysis – Fourier transform frequency

One possibility is the `FourierParameters`

``````  FourierTransform(1, x, w, FourierParameters -> {0, -2*Pi})
``````

``````  FourierTransform(Exp(I a x), x, w, FourierParameters -> {0, -2*Pi})
``````

Compare

``````funs = {1, DiracDelta(x), Exp(I a x), Cos(a x), Sin(a x)};
result= {#, FourierTransform(#, x, w, FourierParameters -> {0, -2*Pi})}& /@ funs;
Prepend(result, {"f(x)","Fourier transform unitary, ordinary frequency"});
Grid(%, Frame -> All)
``````

With wikis second column:

## Fourier analysis – spectral solution of elastic waves

As stated in "Wave Propagation in Structures: an FFT-based Spectral Analysis Method" by James F. Doyle:
For the spectral solution of the 1D equation for elastic waves:
$$widetilde {u} (x, omega) = sum Ce ^ {- i {k_ {n}} x} + De ^ {i {k_ {n}} x},$$
from where $${k_ {n}} = { omega_ {n}} sqrt { rho / E}$$ is the wavenumber. C and D are the indeterminate amplitudes at each frequency. The end of the beam at x = 0 is subjected to a force curve F (t), ie$$EA frac { partial u (x, t)} { partial x} = F (t)$$
E and A are the modulus of elasticity and the cross-sectional area, respectively. The final solution is the inverse Fourier transform of the following expression:

$$widetilde {u} (x, omega) = – frac { widetilde {F_ {n}}} {ik_ {n} EA} e ^ {- ik_ {n} x}$$

$$widetilde {F_ {n}}$$ is the Fourier transform of the applied force F (t).

The numeric example for the above problem is as follows:

``````Rod:
diameter=1 inch
density=0.00247 lb/ci
E=10.6e6 lb/si
Pulse, F(t):
0.000000 0
0.001000 0
0.001100 1000
0.001300 0
0.001500 0
(sec) (N)
``````

I wrote the following code in MatLab:

``````clear all
close all
clc

d=1.0; %inch
A=pi/4*d^2;
rho=0.000247; %lb/inch3
E=10.6e6; %psi
%transform parameters:
n=2^15;
dt=5e-6;
fs=1/dt;
time_fcn = (0:n-1)/fs;
frequency = (0:n-1)*(fs/n);
omega=2*pi*frequency;
F=zeros(1,numel(time_fcn));
nn=find(time_fcn>=0.0011 & time_fcn<=0.0013);
F(nn)=-5e6*(time_fcn(nn)-0.0013);
plot(time_fcn,F)
Fn=fft(F);
plot(omega,Fn)
k=omega*sqrt(0.000247/10.6e6);
A=-Fn(2:numel(omega))./(1i*k(2:numel(omega))*E*A);
x=0;
G(2:numel(omega))= A.*exp(-1i*k(2:numel(omega))*x);
G(1)=simpsons(F,0,max(time_fcn),numel(time_fcn));
U= ifft(G);
plot(time_fcn*1000,U)
``````

The result must be as follows:

However, I can not achieve the same result as stated in the above book. Can someone tell me where my mistake is?

Thank you all,

Greetings.

## Integration – Improper Fourier Transformation

The most common way to check if the Fourier transformation of a function occurs $$f$$ is integrable $$(f in L ^ 1 ( mathbb {R}))$$ is by demonstrating that the function $$f$$ is integrable and at least doubly differentiable. Now I've come across a weaker state where the Fourier transform is not properly integrable. The statement was as follows:

If $$f: mathbb {R} to mathbb {C}$$ is a differentiable, integrable function and even, i. $$f (x) = f (-x)$$, then the Fourier transformation $$has {f}: mathbb {R} to mathbb {C}$$ will be a wrong integral, i. $$has {f} chi _ {(c, d)}$$ is integrable for everyone $$c and the limit $$lim_ {c to – infty} lim_ {d to + infty} int_c ^ d has {f}$$
will exist. In addition, there would be a formula for this impermissible integral in terms of $$f$$?

Can someone give me a hint for this problem? I have no idea how to start?

## Can there be an algorithm faster than the fast Fourier transform to square a polynomial?

FFT is a fast algorithm for multiplying two polynomials, but if it is a square (ie, the polynomial multiplied by itself), can we find something better? (I have reached a paper by Jaewook Chung and M. Anwar Hussain and then something known as the Toom-Cook algorithm, but can no longer find any claims.)

## real analysis – error estimate for the Fourier series of the fraction of \$ x \$

I would like to prove the error estimate for the fractional part of $$x$$ as you can read above.

First, I apply the Abel formula and get
$$left | sum_ {n = N + 1} ^ M frac { sin (2 pi nx)} {n} right | = left | frac { sum_ {n = N + 1} ^ {M} sin (2 pi nx)} {M} + \ sum_ {k = N + 1} ^ {M-1} left ( sum_ {k = 1} ^ n sin (2 pi kx) right) ( frac {1} n- frac {1 } {n + 1}) right |$$

We have that
$$left | sum_ {n = N + 1} ^ M sin (2 pi n x) right | = left | frac { cos (2 pi Nx) – cos (2 pi Mx)} {2 sin ( pi x)} right | = \ left | frac { sin ( pi (N – M) x) sin ( pi (M + N) x)} { sin ( pi x)} right |.$$

I hope my bill is correct. If so, how can I conclude from these equations that (as $$M to infty$$) The mistake is
$$O ((1+ | x | N) ^ {- 1})$$, as $$x$$ approaches any integer and $$N to infty$$?

For me, I think I can see that, though $$x$$ If a fixed number is removed from an integer, then the error is $$O (1 / N)$$, I do not know how to handle it $$x$$ is considered.

## Fourier Transform – Use of FFT in the following convolution in a simulation

I have the following convolution as part of a numerical simulation.

$$T (r) = int d ^ 3r_2 p (r_2) f (r_2) alpha (r-r_2)$$

My problem is that the analytic expressions for $$f$$ and $$p$$ exist however, I have the expression for $$alpha$$ only in the Fourier domain in the form of $$alpha (k)$$, I intended to rate as follows:

1. Construct a grid with the grid of $$100 times100 times100$$ With mesh and linspace in numpy
``````ran = linspace(-1,1,N_r)
x,y,z = meshgrid(ran,ran,ran) #position space
``````
1. Construct the components xf, yf, zf in the Fourier domain from x, y, z
``````xf = fftn(x)
yf = fftn(y)
zf = fftn(z)
``````
1. Find the Fourier transform of $$f (r) times p (r)$$ With FFTn in numpy
2. Multiply it with $$alpha (k)$$
3. Take the inverse Fourier transformation with you ifftn in numpy.

I'm not sure if the above method works, and in fact I could not validate it properly. I've tried using scipy.ndimage.convolve to compare the results with the inverse Fourier transform of the product in the Fourier domain. Is it correct what I do with the code? And is there a way to check if a method works by using a simpler example?

Try to check:

I tried the following to test the theory. Seems like it would not work. I expect the result RES_1 and res_2 be equal. I also used the function real cut off the tiny imaginary part that results from that FFTn and ifftn functions.

``````x = linspace(-1,1,10)
xf = fftn(x)

def f(x):
return x**2+x**3*sin(x)

def g(k):
return k**2+k**3/(3-k**2)

g_k = g(xf)
g_x = real(ifftn(g_k))

res_1 = img_con(g_x,f(x))

res_2 = real(ifftn(g(xf)*fftn(f(x))))

print(res_1)
print(res_2)
``````

Am I doing something wrong?

## pde – heat equation and Fourier series

I try to solve the following boundary value problem with the heat equation:
$$frac { partial {u}} { partial {t}} – frac {1} {4} frac { partial ^ 2 {u}} { partial {x} ^ 2} = 0, ; ; t> 0 text {and} 0 0, \ u (x, 0) = sin {(2 pi x)} – frac {1} {3} sin {(4 pi x)}, ; ; 0
I used the separation of variables $$u (x, t) = X (x) T (t)$$ and went downstairs $$u (x, t) = sum_ {k = 1} ^ { infty} let ^ {- frac {1} {4} (k pi) ^ 2 t} sin {(k pi x) }$$ for a coefficient b.

My question is, how do I use the initial condition? $$u (x, 0)$$ Finish this problem by setting the coefficient & # 39; b & # 39; determine. I'm assuming that I would have to use Fourier series for sine, but I'm not sure how to handle it. Every help is appreciated.