Signal Processing – Graph Fourier Transform: the adjoint notation for the eigenbase matrix

It is known that for a true symmetric matrix $ L $ (here Graph Laplace) one can write the eigenvalue decomposition as

$$
L = U lambda U ^ { mathsf T},
$$

from where $ U $ is a real eigenvector matrix,
In graph signal processing papers, including the grand paper of Shuman et al. (see page 4), the adjunct (complex conjugate) of $ U $ is used to define the graph Fourier transform $ mathcal {F} _ {G} $ as
$$
has {x} = mathcal {F} _ {G} x = U ^ {*} x,
$$

from where $ x $ is the signal in vector form and $ U ^ {*} $ is the complex conjugate of $ U $,

I am curious if there is a specific reason for using the notation of complex conjugate?

Discrete Fourier transform of $ (1,1,1,1) $

I am asked to determine the Fourier transform of $ (1,1,1,1) $,

In the solution I found the following:
solution

I do not understand how he got from the $ omega $is to $ -i, i, -1, 1 $ etc…

How can you break it down to make it easier to understand? Which middle step (which could better explain the way to the solution) does not the professor write?

Any hints?

Characterization of a modular form over its first Fourier coefficients at infinity

It is known that a hump form
$$
f = sum_ {n ge 1} a_n q ^ n
$$

from the weight $ k $ and level $ 1 $ is determined by its first $ d_k = text {dim} S_k $ Coefficients. This follows from the valence formula (which gives a limit) and an explicit construction using Eisenstein series.

What happens at higher levels, for example for the group? $ Gamma_0 (N) $? The valence formula still gives a limit, but can we pinpoint the smallest integer? $ d_ {k, N} $ so that $ f in S_ {k} ( Gamma_0 (N)) $ disappears exactly when $ a_n = 0 $ to the $ 1 le n le d_ {k, N} $?

What is known and what is suspected?

Fourier analysis – squaring of complexes in the wavenumber domain

Suppose I have 2 vectors of complex numbers in the wavenumber range: $ A $ and $ B $,

After inverse Fourier transformations, their corresponding counterparts are the space domain: $ a $ and $ b $,

I would like to calculate an example of a time domain result with an exponent term:
$ c = ab ^ 2 + (a-b) ^ 2 $

For this I can easily perform separate inverse Fourier transformations $ A $ and $ B $, before the count $ c $, However, this is very time consuming as my number of variables increases because a Fourier transform is required for each variable.

Instead, it would be more efficient to calculate: $ C = AB ^ 2 + (A-B) ^ 2 $ directly in the wavenumber domain, and then perform only a single Fourier transform of $ C $ to restore the time response, $ c $, However, this is a challenge because the quadratic term causes problems due to the presence of complex numbers. How can I overcome this?

Function analysis – Fourier transformation of the generator operator on the Schwartz spcae

To let $ f in S ( mathbb {R}) $, from where $ S ( mathbb {R}) $ denotes the Schwartz space,
and $ A = frac {1} { sqrt {2}} (x + frac { text {d}} { text {d} x}) $ be the creation operator and $ A ^ dagger = frac {1} { sqrt {2}} (x- frac { text {d}} { text {d} x}) $ the annihilation operator.

Now I have to show the following:

(on) $ mathcal {F}[Af]= iA mathcal {F}[f]$

(B) $ mathcal {F}[A^dagger f]= -iA ^ dagger mathcal {F}[f]$

and C) $ mathcal {F}[Hf]= H mathcal {F}[f]$ With $ H = A ^ Dagger A $,

Here $ mathcal {F} $ represents the Fourier transform.

I've plugged in the definitions and tried to integrate them into parts, but I think I'm missing something. Help is so much appreciated!

Calculus – Calculation of the Fourier coefficients with Excel

I'm trying to use Excel to compute Fourier coefficients.
But I'm confused which method should be used to approximate int (f (x) cos (x) dx) and int (f (x) sin (x) dx).

1st trapezoid * cos (x) & trapezoid * sin (x)
2. Trapezium * specific integral (cos (x) dx) & trapezoid * specific integral (sin (x) dx)

I have found a sample excel sheet (https://quickfield.com/harmonics_analysis_excel.htm) that seems to have the particular integral of the cosine and sine term.

Is this a more accurate method than the normal averaging of f (x) cos (x) and f (x) sin (x)?

How can I express a picture with epicycles with Fourier series?

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