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)
```

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# Tag: Fourier

## Inverse Fourier Sequence Transform returns zero

## How does the Fourier series relate to orthogonality?

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

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

## Fourier analysis – Determination of frequency components using FFT

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

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

## Fourier analysis – Fourier transform frequency

## Fourier analysis – spectral solution of elastic waves

## Integration – Improper Fourier Transformation

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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)
```

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?

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?

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
def polarToRectangular(radii, angles):
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

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))
```

To let $ Y $ Let be the set of integers $ N <n leq 2 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?

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.

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:

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.

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 <d $ 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?

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