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

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?

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.

left: sine signal + noise, right: sine frequencies

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.

Thank you in advance.