Fourier series – I need help to finish and draw this EDP

I have an EDP that I digitally solve on a 2D polar annular grid. I am trying to solve the axisymmetric analysis analytically in order to test my numerical solution.

The PDE:

$$ frac {1} {s} frac {} {∂s} ( frac {s} {ρ} frac {∂ψ} {} s}) + frac {1} {s ^ 2} ⋅ frac {∂} {} ( frac {1} {ρ} frac {∂ψ} {∂Φ}) – 2Ω + ρc_0 + ρc_1ψ = 0 $$

Or
$$ Ω, ρ, c_0, c_1 $$
are known constants (ρ is constant only in the axisymmetric case, normally depends on s and

Using the separation of variables, I think we get:

$$ = frac {1} {ρc_1} (AJ_k (ρs sqrt {c_1}) + BY_k (ρs sqrt {c_1})) (Ccos (kΦ) + Dsin (kΦ)) + frac {2Ω} { ρc_1} + frac {c_0} {c_1} $$

I do not know what to do here. I want to apply the boundary conditions of Dirichlet to an internal limit s0 and an external limit s1. Are these solutions in series? As the summations on k = 1,2 … and if so, how can I handle the coefficients (A, B, C, D)?

My instinct is to say that k = 0 for the dependent part to become a constant, but I'm not sure how to handle the bessel functions as this would limit them to only bessel functions of order 0. I am generally little experienced in the use of Bessel functions.

How should this problem be solved with only 2 boundary conditions (inner and outer rings of the domain)? I do not even try to apply neumann conditions yet.

Thank you!

Fourier Analysis – Explicit Limits of Tao's Result on Collatz's Conjecture

A new pre-print of Terry Tao has been published recently and has yielded interesting results on the topic of Collatz's conjecture. I will not cite the precise result, but rather an equivalent formulation that Tao notes in his remark 1.4:

For all $ delta> $ 0 there is a constant $ C_ delta $ such as $ mathrm {Col_ {min}} (N) leq C_ delta $ for everyone $ N $ in a subset of $ mathbb N + 1 $ at least logarithmic density $ 1- delta $.

My question is something about the growth rate of $ C_ delta $ as $ delta to 0 $. I will ask two specific questions in this regard. The first, I imagine, could have been known before the recent Tao result.

Are there values ​​of $ delta <1 $ for which a explicit upper limit for $ C_ delta $ is known?

The second essentially asks if something explicit can be deduced from the result of Tao.

Is the function $ N mapsto C_ {1 / N} $ Upper bound by a computable function?

(Note that I do not think the answer is "obviously yes" to the mere existence of $ C_ delta $since the control of the operation of a particular value seems to be $ Sigma ^ 0_3 $ (it exists $ N_0 $ as for all $ N> N_0 $ it exists $ M $ as in $ M $ not at least $ 1- delta $ numbers below $ N $ to go downstairs $ C_ delta $) and is obviously not lower)

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

Let $ A: = {u in H ^ k (0,2 pi): u ^ {(j)} (0) = u ^ {(j)} (2 pi) mbox {for} j = 0.1, ldots, k-1 } $, or $ H ^ {k} (0,2 pi) subseteq L ^ 2 (0, 2 pi) $ is the space of the order Sobolev k $ sure $ (0, 2 pi) $. Can we say that $ u in A $ Yes Yes
$$ sum_ {n = – infty} ^ infty (1 + n ^ 2) ^ k | hat {u} (n) | ^ 2 < infty? $$
In the series above $ hat {u} (n) $ are the Fourier coefficients of $ u $. We think the answer to this question is affirmative because maybe we can identify $ A $ with the space of Sobolev of the torus $ H ^ k ( mathbb {T}) $and use this result. But we do not know how to show that there is an isomorphism between $ A $ and $ H ^ k ( mathbb {T}) $.

Do you know a reference for a characterization of $ A $ with the Fourier series?

Thank you for any help you can provide us.

Fourier Analysis – FourierTransform Frequency

One of the methods is to adjust the FourierParameters

  FourierTransform(1, x, w, FourierParameters -> {0, -2*Pi})

Mathematica graphics

  FourierTransform(Exp(I a x), x, w, FourierParameters -> {0, -2*Pi})

Mathematica graphics

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)

Mathematica graphics

With the second column of Wiki:

Mathematica graphics

Fourier analysis – Jordan's theorem as a generalization of Dirichlet's theorem

A well-known result (Jordan's theorem) indicates that the Fourier theory of a 2pi-periodic function converges to the value of the function at points where the function is continuous, provided that the function has a bounded variation.

This is supposed to be a generalization of Dirichlet's theorem, which says that continuously differentiable 2pi-periodic functions (possibly except on a finite set) have this property. However, I do not understand how generalization works.

Here's what I know:
If the Fourier coefficient of a function decreases as fast or faster as 1 / n, there is a convergence towards the value of the function at the points of continuity.
If we have a continuous function (except possibly on a finite set) of bounded variation, the Fourier coefficient will decrease faster (or as fast as) 1 / n.

So, if Jordan's theorem enunciated the result for continuous functions (except for many points) with bounded variation, I would understand where it came from. But the result is indicated for ALL functions of bounded variation. Where does this come from?

Finite Fourier Transformation

I have a problem on the next question. I can easily prove it $ max_j | hat f (j) | geq sqrt {n} $ using the formula $ sum_j | f (j) | ^ 2 = frac {1} {n} sum_j | hat f (j) | ^ 2 $ but I do not know how to exclude equality.

Problem: $ f in l ^ 2 (Z_n) $and suppose that $ | f | = X_E, E subset Z_n $. Suppose the support of $ hat f $ is contained in a set of size at most $ | E | $. CA watch $ max_j | hat f (j) |> sqrt {n}. $

Numerical Methods – Elliptical Fourier Adjustment Coefficients

I'm trying to implement a function that could correspond to an elliptic Fourier curve on a set of border points of a detected object. I use cv2.findContours to acquire border points from a binary image. Then I would like to calculate the elliptic Fourier coefficients using the equation:

(for the sake of simplicity, I will only discuss the x-axis)

here is the equation:

the coefficients

And here is my question:
The idea is to set the x coordinates from 0 to 2 * π. My question is: if Δt should be a constant or should it depend on Δx (the greater the change in x-coordinate, the larger the Δt)?

sound – Fourier analysis of a .aiff file

I'm trying to use Mathematica to perform Fourier analyzes on audio files, but I do not know how to get the frequency spectrum from the .aif file.

The instructions I have said:

  1. use the Mathematica Read command to take samples of the tone data at 11,025
    samples per second.

  2. use the Export command in Mathematica to convert the data to an .aif file.

  3. use GarageBand to edit the sound wave data with any of the 15 effects.

  4. use the Import, Fullform, and Table commands in Mathematica to convert the
    new .aif file in 11,025 data samples.

  5. use the Mathematica Fourier and ListLinePlot commands to generate and
    plot the frequency spectrum of the modified sound wave data.

Therefore, I first produced a pure sound of one second while writing Play[Sin[2 Pi 440 t], {t, 0, 1}], naming it & # 39; test & # 39; and exporting it with Export["test.aiff", test]

When I get stuck, I convert the .aif file back to data points using the Fullform and Table commands, and convert the Fourier transform into a frequency spectrum with the help of ListLinePlot.

If anyone could help, that would be greatly appreciated, thanks in advance

Why is the passing limit due to the Gibb phenomenon not zero if a Fourier series with infinite terms is not exceeded?

I'm not very familiar with the operation of boundaries at a deeper level, but is there an intuitive reason why:

  • The limit of the passing of a Fourier approximation as $ m to infty $ is about 0.0895. $ I saw this for partial sums of the Fourier approximation for a function with a jump discontinuity.
  • But I also heard that $ m to infty $ the fourier series that you get should be an exact replica of the function that it imitates (that is, without exceeding)?

So how is it that the error / overshoot does not seem to disappear because of the first point, but does it seem to disappear because of the second point?

Fourier series, parametric curve of an image using Fourier transformation

I did these actions:
img = Import("https://www.clipartmax.com/png/middle/100-1003682_homer2-homer-simpson-crazy-png.png");
img = Binarize(img~ColorConvert~"Grayscale"~ImageResize~500~Blur~2);
pts = DeleteDuplicates@Cases(Normal@ListContourPlot(Reverse@ImageData(img), Contours -> {0.5}), _Line, -1)((1, 1))

and got a table of points. How can I get a parametric equation (x (t), y (t)) of these points, using Fourier, FourierTrigSeries? I want to get equations like x(t) = 245.196 + 121.653 Cos(t) + 17.6594 Cos(2 t), y(t) = 347.468 - 202.673 Cos(t) - 26.0902 Cos(2 t) - 12.7999 Cos(3 t) -
6.15289 Cos(4 t) + 4.381 Cos(5 t)
with a given precision