## Inverse Fourier Sequence Transform returns zero

I wonder why Mathematica (12) in the following code

``````InverseFourierSequenceTransform(1,kx,x)
``````

gives different results if we change only "1" to "1".

``````InverseFourierSequenceTransform(1.,kx,x)
``````

## How does the Fourier series relate to orthogonality?

It's hard for me to see the relationship between the two. I watched videos about Fourier series and the various animations. I do not understand how this relates to orthogonality. Can someone help me to understand the relationship?

## Fourier series of a function in the interval [-pi/2,pi/2]

I have to show that $$frac { cos x} {3} + frac { cos 3 x} {1 cdot 3 cdot 5} – frac { cos 5 x} {3 cdot 5 cdot 7} + frac { cos 7 x} {5 cdot 7 cdot 9} – cdots = frac { pi} {8} cos ^ {2} x$$ on the interval $$(- pi / 2, pi / 2)$$, I tried to find the Fourier coefficient $$a_n$$ with the formula $$a_n = (2 / ( pi / 2)) int_0 ^ { pi / 2} ( pi / 8) cos ^ 2 (x) cdot cos (k pi x / ( pi / 2 ))$$, but in the end always something useless. How do I do it?

## Implementation of 3Blue1Browns Description of the Fourier transform in Python + numpy

I have implemented the description of the 3Blue1Brown Fourier transform in Python + numpy for irregular and unsorted data as described here.

``````import numpy as np
import matplotlib.pyplot as plt

return radii * np.exp(1j * angles)

def frequencyGenerator(time, steps = 100):
𝛿 = time.max() - time.min()
M = np.arange(1, steps + 1)(:, np.newaxis)
return M / 𝛿

def easyFourierTransform(time, values, frequency = None, steps = 100):
if frequency is None:
ft = frequencyGenerator(time, steps)
frequency = ft.reshape(ft.shape(0))
else:
ft = frequency(:, np.newaxis)

# sorting the inputs
order = np.argsort(time)
ts = np.array(time)(order)
Xs = np.array(values)(order)

𝜃 = (ts - time.min()) * 2 * np.pi * ft
Y = polarToRectangular(Xs, 𝜃)(:, :-1) * np.diff(ts)
amplitude = np.abs(Y.sum(axis=1))
return frequency, amplitude
``````

I think maybe I can suggest adding this to numpy / scypy. However, I am not sure if this little piece of code is qualified for upstream addition. I was wondering if you could help me to know the following:

• Is this code correct? Is there indeed the Fourier transform back? I want to make sure that there are no logical errors.
• Is it a high-performance code or is there a way to improve performance?
• Is formatting good enough? Should I comply with the PEP8 standard or require Numpy / Scipy other best practices?
• How can I add a write comment? In particular to make sure that the dimensions of the ndarrays are in order.
• Is this novel or has it been written before? (not necessarily relevant to this forum, but still my question)

I would be glad if you could help me with the above mentioned points. Thank you in advance.

Keywords: uneven, even, uneven, sampled, distributed

## 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?

Thank you for any help you can give us.

## 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?