## reference request – Energy estimation for linear hyperbolic system (without Fourier)

Where can I find proof of an energy estimate (under appropriate assumptions) for the following linear hyperbolic system which is not based on Fourier methods?

$$frac { partial U} { partial t} + sum_ {j = 1} ^ m A_j frac { partial U} { partial x_j} + B cdot U = 0,$$
or $$x in mathbb R ^ m$$ and $$U in mathbb R ^ n$$ and $$A$$ is a symmetric matrix.

## Solving a system and taking inverse Laplace / Fourier transforms

I have a set of linear equations for 4 quantities that have been Fourier and Laplace transforms. The system must be solved for the quantities and then each of the quantities must be transformed into an inverse Laplace, then transformed into an inverse Fourier (where the transforms are all one-dimensional so there is only one frequency).

I tried to solve the system with Maple, then I use the invlaplace function to take the Laplace inverse transforms of the quantities. The code I use is as follows:

e1: = -2 * DIkpi + As = 0

e2: = 2 * AIk * pi + 2 * CIkpi + Bs = a

e3: = s * C + 4/5 * Pw3(2 * ftIk * D – 2 * 1/3 * piIkD) / w2 = -2(C + 2 * K * (2 * piIk * B – 2 * 1/3 * piIk * B)) / (w2 * K)

e4: = s * D + 2 * 5/4 * P * t4 * piIk * C / (t2 * K) = -5 / (2 * P) * D / (t2 * K)

sys: = {s * C + 4/5 * Pw3(2 * ftIk * D – 2 * 1/3 * piIkD) / w2 = -2(C + 2 * K * (2 * piIk * B – 2 * 1/3 * piIk * B)) / (w2 * K), s * D + 2 * 5/4 * P * t4 * piIk * C / (t2 * K) = -5 / (2 * P) * D / (t2 * K), -2 * DIkpi + As = 0.2 * AIk * pi + 2 * CIkpi + Bs = a}

solve (sys, (A, B, C, D)),

where everything is a constant apart from the frequency $$k$$, the variable $$s$$ is from Laplace's transformation and $$i = sqrt {-1}$$. The solution Maple produces for the 4 quantities is a fairly complicated fraction, and when I use invlaplace on each quantity, I get the answer as a sum on the roots of a quartic polynomial.

I was wondering if it would be possible to use Mathematica to solve the system and get the quantities in the simplest form possible, then if there is a function similar to invlaplace which then allows me to Get a relatively simple analytical expression for the inverse 1D Laplace transformation for each of the magnitudes, so that they can be passed to a function that gives the inverse Fourier transform 1D?

## ag.algebraic geometry – Why is the Fourier transform so ubiquitous?

Many operations and equivalences in mathematics appear as a kind of Fourier transform. By Fourier transform, I mean the following:

Let $$X$$ and $$Y$$ be two objects of a certain category with products, and consider correspondence $$X leftarrow X times Y to Y$$. If we have an object (think of the sheaf, the function, the space, etc.) $$mathcal {P}$$ more than $$X times Y$$ and another, let's say $$mathcal {F}$$ more than $$X$$, assuming the existence of proper push and pull operations, we can get another object on $$Y$$ pulling $$mathcal {F}$$ back to product, tensor with $$mathcal {P}$$, then pushing towards $$Y$$.

The standard example is the Fourier transform of functions on a locally compact abelian group $$G$$ (for example. $$mathbb {R}$$). In that case, $$Y$$ is the Pontryagin dual of $$G$$, $$mathcal {P}$$ is the exponential function on the product, and the push and pull are given by integration and precomposition, respectively.

We also have the Fourier-Mukai functors for coherent showers in algebraic geometry which provide the equivalence of coherent showers on double abelian varieties. In fact, almost all of the interesting functors between coherent bundles on fairly pleasant varieties are examples of Fourier-Mukai transforms. A variant of this example also provides the geometric Langlands correspondence

$$D (Bun_T (C)) simeq QCoh (LocSys_1 (C))$$

for a torus $$T$$ and a curve $$C$$. In fact, the Langlands geometric correspondence for general reducing groups also seems to come from such a transformation.

Speak $$SYZ$$ guess, two Calabi-Yau mirror collectors $$X$$ and $$Y$$ are Lagrangian double-core fibrations. As such, conjectured equivalence

$$D (Coh (X)) simeq Fuk (Y)$$

is obtained morally by applying a Fourier-Mukai transform which ignites coherent sheaves $$X$$ in Lagrangiens in $$Y$$.

To make things more mysterious, many of these examples are the result of the existence of a perfect match. For example, the bundle of Poincaré lines which provides the equivalence of coherent sheaves on double abelian varieties $$A$$ and $$A ^ *$$ results from the perfect combination

$$A times A ^ * à B mathbb {G} _m.$$

Likewise, the Langlands geometric correspondence for the toroids, as well as the GLC for the Hitchin system, arise in a way from the self-duality of the Picard stack of the underlying curve. These examples seem to show that the non-degenerate quadratic forms seem to be fundamental in a very deep sense (for example perhaps even the Poincaré duality could be considered as a Fourier transform).

I don't have a specific question, but I would like to know why we should expect Fourier transformations to be so fundamental. These transformations are also found in physics as well as in many other "real world" situations of which I am even less qualified than my examples above. Nevertheless, I have the feeling that something deep is going on here and I would like an explanation, even a philosophical one, on why this model seems to appear everywhere.

## Fourier analysis – Reference request: explicit construction of Kakeya sets using Perron's tree

I have found many excellent notes online that illustrate how to build a set of Kakeya needles (with measure $$< varepsilon$$.) Yet none of them gives a complete argument on the construction of a set of Kakeya (with zero measure). The closest is given on page 6 of

https://web.stanford.edu/~yuvalwig/math/teaching/KakeyaNotes.pdf,

which unfortunately leaves the detail to argue for the existence of a unitary line segment in $$cap_ {i = 1} ^ infty U_n$$. He says this is proven subtly using a compactness argument.

What is this argument?

## Help verifying a Fourier transformation identity

I don't know why I'm getting stuck, but I'm having trouble verifying the following identity in one of my textbooks:

$$chi (T ( lambda – lambda_j)) = frac {1} {T pi} int _ {- infty} ^ { infty} hat { chi} (t / T) e ^ {it lambda} cos (t lambda_j) , dt + chi (T ( lambda + lambda_j))$$

I know I have to separate the complex exponential and use the Euler identity, but I'm having trouble getting the $$chi (T ( lambda + lambda_j))$$ term to work. My calculations keep coming with $$chi (T ( lambda_j – lambda))$$.

I don't think it's important for the calculation, but $$T> 1$$, $$lambda, lambda_j geq 0$$, and $$chi$$ is a function of the Schwartz class.

## derived categories – Fourier Mukai kernel which gives equivalence in one direction

Yes $$X$$ and $$Y$$ are two schemes and $$F in Perf (X times Y)$$, we can then define a functor $$Perf (X)$$ at $$Perf (Y)$$ like the Fourier Mukai transformation $$Phi ^ {X rightarrow Y} = q _ { ast} (F otimes p ^ { ast} -)$$, or $$p$$ and $$q$$ are the projections of $$X times Y$$ at $$X$$ and $$Y$$ respectively. Yes $$X$$ and $$Y$$ are smooth we can change $$Perf (-)$$ at $$D ^ b (-)$$, while if you choose $$F$$ to be a complex with almost coherent cohomologies, we change $$Perf (-)$$ at $$D_ {qc} (-)$$, the unbounded derived category of complexes with almost coherent cohomologies. Obviously, we could exchange the role of $$p$$ and $$q$$ and get a functor of $$Y$$ at $$X$$. My question is whether the fact that $$Phi ^ {X rightarrow Y}$$ 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 this might not be the case that the implication holds is that there is no apparent relationship between the two functors that I have described above. on it, even if they share the same core. However, I couldn't find a counterexample.

## What is the physical meaning of this property of the Fourier transform?

What is the physical meaning of this property of the Fourier transform?

$$int _ {- ∞} ^ {+ ∞} （f (x)） ^ 2 dx = frac1 {2π} int _ {- ∞} ^ {+ ∞} | F (ω) | ^ 2 dω,$$ ($$F (ω)$$ is the Fourier transform of $$f (x)$$)

The Fourier transform is widely used in electricity. Explanations in this or other areas are all welcome. A reference in the relevant articles is appreciated!

By the way, how do you type a formula in the text box when you ask a question? Thank you!

## Laplace and Fourier transform to derive an unstable fundamental solution

I consider the equations of the unsteady Stokes flux

$$nabla cdot u = 0,$$

$$rho frac { partial u} { partial t} = – nabla p + mu nabla ^ 2 u.$$

In an article (which I can provide if necessary), it is stated that one can use Fourier and Laplace transforms on these equations to obtain the unsteady fundamental solution.

I haven't used a lot of transformation theory before, so I was wondering how it would work, do we separate the variables to have an ODE in the time variable which is solved with the Laplace transform and the Fourier transform is preserved for the spatial part?

## Fourier analysis – Scaling of a Mexican hat wavelet into continuous wavelet transform

My general question is related to the concept of scaling a wavelet whose analytical form exists. An example is that of a Mexican hat.

The Mexican hat wavelet, obtained from the second derivative of a Gaussian, has a functional form of 2 * (1- (t / s). ^ 2). * Exp (-t. ^ 2 * 0.5 / s ^ 2) divided by (pi) ^ (1/4) * sqrt (3 * s), where "s" is the standard deviation of ; a Gaussian and $$t$$ is the independent variable.

If we want to graphically see its compressions and dilations graphically, let's say on a scale $$a$$ = 1 to 5, the typical form of a wavelet function $$psi$$ has a scale setting $$a$$ and a translation parameter $$b$$. The functional form of the Mexican hat given in MATLAB does not explicitly have them both.

Standard deviation $$s$$ is the only variable here that controls the width, but this is not strictly equal to the scale $$a$$. How to generate scale versions of the Mexican hat?

Thank you.

P.S. I posted this digital signal processing, but received no response.

## Fourier analysis – Reference (fundamental soil and grad estimation, etc.): a particular elliptical PDE

in $$mathbb {R} ^ d$$, consider the following equation
$$Delta u -x cdot nabla u = f$$
or $$f$$ may be $$C ^ infty$$ and exponentially rapid decay.

I would like to know the fundamental soil. to this equation or to a gradient estimate, preferably punctually. In particular I am interested in the following quantity
begin {align *} int _ { mathbb {R} ^ d} | nabla u | ^ 2e ^ {- frac {| x | ^ 2} {2}} dx. end {align *}

Thank you!