## Fourier Analysis – Show no lower limit for the vibration integral

Consider the vibration integral
$$I (x) = int_0 ^ 1 e ^ {ix t ^ 2} dt.$$
The standard Van der Corput lemma tells us that $$I (x) = O (| x | ^ {- 1/2})$$ to the $$| x |> 1$$,

I was looking for a lower limit like this: $$| I (x) | geq c | x | ^ {- 1/2}$$, but it turns out to be impossible because someone tells me $$I (x)$$ has infinite zeros $$x_n to infty$$, But is it true? If so, how can I prove it?

## Advantage of the fractionated Fourier transform over multiscale wavelet

What is the best argument for a fractional Fourier transform over a multiscale wavelet for data analysis purposes?

_ Optimization of the good time-frequency domain parameter? "Good" is to find the% time% fq domain that minimizes the spectral entropy of the data.

_ Behavior in the real physical world, since the particle path is described locally by fractional Fourier in the N-gap problem context?

_ Wavelet is just a quick log2 implementation of FractFt?

_ Continuous transformation from time to frequency … But for what purpose?

I take every strong argument from fracFt about Wavelet

## Fourier expansion of the inner product mod 2

I am trying to read Ryan O & # 39; Donnell's book "Analysis of Boolean Functions" and am going to repeat this part of an exercise:

• Calculate the Fourier extension of the indicator function mod 2 $$mathrm {IP} _ {2n}: mathbb {F} _2 ^ n rightarrow {- 1,1 }$$, defined by $$mathrm {IP} _ {2n} (x_1, ldots, x_n, y_1, ldots, y_n) = (-1) ^ {x cdot y}$$

Any tips?

## Calculus – What is the Fourier transform in spherical coordinates?

Assume a function in spherical coordinates $$f (r, phi, theta)$$, Ask about the Fourier transform of such a function.

Suppose the function $$f$$ was symmetrical in relation $$phi, theta$$i.e. $$f (r, , ) = f ()$$,
Then what is the spherical coordinate Fourier transform for such a function?

There is the usual definition $$int_ {0} ^ infty f (r, phi, theta) r ^ 2dr int_0 ^ {2 pi} d phi int_0 ^ pi d theta e ^ {- i vec {p } cdot vec {x}}$$ didn't seem to apply entirely. Since $$vec {x} = (r, phi, theta)$$ and $$vec {p} = (p_r, p_ phi, p_ theta)$$, The latter didn't even seem to be properly defined. Even if you treat it as a one-dimensional case, the coefficient will be $$frac {1} { sqrt {2 pi} ^ 3}$$ or just $$frac {1} { sqrt {2 pi}}$$?

How is the Fourier transformation carried out in spherical coordinates?

## The Gimp Fourier plugin cannot be used on Ubuntu 18.04.4

I tried to install this fft plugin for GIMP:

GIMP Plugins

following:

``````tar xvzf fourier-0.4.*.tar.gz
cd fourier-0.4.*
make clean
make
make install
``````

After that, I added the plugin folder (created by the script to copy the compiled plugin, i.e. `/home//.gimp-2.8/plug-ins/` ) in the GIMP plugin folder settings.

## Fourier transform norm – mathematical stack exchange

Let's say we have one $$W in C ^ n$$ Signal and a $$W ’= Omega_n W$$ as a discrete Fourier transform of $$W$$,

Is the norm of $$W$$ same norm of $$W ’$$? (||$$W$$|| = ||$$W ’$$||)

And the second question can $$W ’$$ used to determine how much specific frequencies of $$W$$ oscillate?

## Certain Integrals – Solve this Fourier transform that is too long for the title …

I want to solve the following integral:

$$f_ {X} (x) = int _ {- infty} ^ { infty} frac {e ^ { frac {-j (N-1) alpha t – (N + 1) alpha beta t ^ {2}} {(1-j beta t) (1 + j beta t)}} {(1-j beta t) ^ {N} (1 + j beta t) ^ { N}} e ^ {jxt} dt$$

Where $$alpha, beta in mathbb {R} _ {+}$$ and $$N in mathbb {Z} _ {++}$$ and $$j = sqrt {-1}$$,

However, I'm not sure how to deal with it $$t ^ {2}$$ in exponential.

It seems to indicate completing the square, but maybe I'm failing in my algebra because I'm stuck.

** The completion of the square idea: **

We extend the integrand as follows:

$$e ^ { frac {-j (N-1) alpha t} {(1-j beta t) (1 + j beta t)}} e ^ { frac {- (N + 1) alpha beta t ^ {2}} {(1-j beta t) (1 + j beta t)}} e ^ { frac {jxt (1-j beta t) (1 + j beta t)} {(1-j beta t) (1 + j beta t)}}$$

Then I try to eliminate all imaginary numbers except the coefficients of $$xt$$So we can do a Fourier transform of a rational function and use the resideu thm. But I'm stuck here … maybe this is not the right approach.

Anyway … Mathematica doesn't seem to have found an answer.

Any ideas?

## derived categories – Fourier Mukai kernel, which gives equivalency only in one direction

If $$X$$ and $$Y$$ are two schemes and $$F in Perf (X times Y)$$Then we can define a functor from $$Perf (X)$$ to $$Perf (Y)$$ how to transform the Fourier Mukai $$Phi ^ {X rightarrow Y} = q _ { ast} (F otimes p ^ { ast} -)$$, Where $$p$$ and $$q$$ are the projections of $$X times Y$$ to $$X$$ and $$Y$$ respectively. If $$X$$ and $$Y$$ are smooth we can change $$Perf (-)$$ to $$D ^ b (-)$$while if you choose $$F$$ we only change to be complex with quasi-coherent cohomologies $$Perf (-)$$ to $$D_ {qc} (-)$$, the indefinitely derived category of complexes with quasi-coherent cohomologies. Obviously we could switch roles from $$p$$ and $$q$$ and get a functor from $$Y$$ to $$X$$, My question is whether this is so $$Phi ^ {X rightarrow Y}$$ is an equivalence implies that $$Phi ^ {Y rightarrow X}$$ is an equivalence, and if not, I would like a counterexample.

The reason why I think that this might not be the case is that there is no obvious relationship between the two functors described above, even though they share the same kernel. I couldn't come up with a counterexample.

## How can I calculate sine signal frequencies from a sine + noise signal and its Fourier transform graphs?

I want to calculate the frequency of the two sinus peaks in the graphic below, but I really don't know how to do it.

So far I only know that the signal is $$f (k) = sin (k) + n (k)$$ The description shows that the length of the signal is 255 seconds with 255 discrete values. How can I calculate the peaks from the right side (i.e. the frequency of this sine wave should be a little under 50, measured on the left peak)?

Incidentally, the frequencies on the right are changed to (0.255) instead of (-128.127) to avoid confusion

## Joint optimization of the order 1 moment of a function and its Fourier transformation

For a quantum optics experiment I come across the following problem:

To let $$X, P in mathbf {R} ^ 2$$

$$begin {equation} J ( psi) = | int _ {- X} ^ {X} x | psi ^ 2 (x) | dx + int _ {- P} ^ {P} p | Tf ( psi) ^ 2 (p) | dp | end {equation}$$

Find:

begin {equation} Left{ begin {align} sup_ { psi} J ( psi) \ text {with restriction} int _ { mathbf {R}} | psi ^ 2 (x) | dx = 1 end {align} right. end {equation}

where Tf is the Fourier transform

$$psi$$ functions should be fluid enough for every purpose. Therefore I do not explicitly say in which room they live.

I tried it like this:

1) If I refer every term upwards independentlyI can get X + P with a $$psi ^ 2 (x) = Delta (X)$$ and $$Tf ( psi ^ 2 (p)) = Delta (p)$$ , From now on, the goal is to make the most of the relationship between the two $$psi$$ and his Tf

2) In quantum physics, the problem can be described in the following form:

$$begin {equation} J ( psi) = | int _ {- X} ^ {X} x | psi ^ 2 (x) | dx + int _ {- P} ^ {P} psi ^ * (x) frac { partially psi} { partial x} – psi (x) frac { partial psi ^ *} { partial x} dx | end {equation}$$

I tried to distinguish J in terms of $$psi$$ and $$psi ^ *$$ to apply Lagrangian multipliers to the functional space of $$psi$$ functions. With X = P, I found:

$$begin {equation} exists lambda \ text {so that} \ 2 i frac { partial psi} {x} + (x – lambda) psi = 0 end {equation}$$
what I solved with
$$begin {equation} psi = C exp left ( frac {i (x ^ 2 – lambda x)} {2} right) end {equation}$$

Could you please help me with the general problem? Would you know that literature treats it? It seems to me that this is related to the Pauli problem, but a little differently.