fourier analysis – Inclusion map $i : C^{0,alpha}[0, 1] rightarrow C^{0,beta}[0, 1] $ is linear, and therefore compact

Q) Show that the inclusion map $i : C^{0,alpha}(0, 1) rightarrow C^{0,beta}(0, 1) $ is compact.

Ans) Let {$u_n$}$_1^infty$$C^{0,β}(0,1)$ such that $|u_n|_{C^{0,β}} leq 1$ , i.e. $|u_n|_infty leq 1$ and
$$|u_n(x) − u_n(y)| leq |x − y|^beta
text{ for all } x, y ∈ (0,1)$$
By Arzela-Ascoli, there exists a subsequence {$tilde u_n$}$_1^infty$ of {$u_n$}$_1^infty$ and $u ∈ C(0,1)$ such that $tilde u_n rightarrow u$ in $C$. Since $$|u(x) − u(y)| = lim_{n→∞} |tilde u_n(x) − tilde u_n(y)| ≤ |x − y|
β$$

$u ∈ C^{0,β}$ as well. Define $g_n := u − tilde u_n ∈ C^{0,β}$, then
$$(g_n)_{0,β} + |g_n|_C = |g_n|_{C^{0,β}} ≤ 2$$
and $g_n → 0$ in $C$. To finish the proof we must show that $g_n → 0$ in $C^{0,α}$. Given
$δ > 0$,
$$(g_n)_{0,α} = sup_{substack{x,yin (0,1) \ x neq y}} frac{|g_n(x)-g_n(y)|}{|x-y|^{alpha}} ≤ A_n + B_n$$
where
$$A_n = sup Biggr { frac {|g_n(x) − g_n(y)|}{|x − y|^α} : x neq y text{ and } |x − y| ≤ δ Biggr }$$ $$= sup Biggr { frac {|g_n(x) − g_n(y)|}{|x − y|^β}·|x − y|^{β−α} : x neq y text{ and } |x − y| ≤ δ Biggr }$$ $$≤ δ^{β−α}·(g_n)_{0,β} ≤ 2δ^{β−α}$$
and
$$B_n = sup Biggr { frac {|g_n(x) − g_n(y)|}{|x − y|^α} : |x − y| > δ Biggr }$$
$$≤ 2^{δ−α}|g_n|_C → 0 text{ as } n → ∞$$
Therefore,
$$limsup_{n→∞}
(g_n)_{0,α} ≤ limsup_{n→∞}
A_n + limsup_{n→∞}
B_n ≤ 2δ^{β−α} + 0 → 0 text{ as } δ ↓ 0$$

$ $

Now the Definition of compactness for the map being used is

Let $X$ and $Y$ be normed spaces and $T : X → Y$ a linear operator. Then $T$ is compact if for any bounded sequence $(x_n)_{n in mathbb {N}}$ in X, the sequence $(Tx_n)_{n in mathbb {N}}$ contains a converging subsequence

Assuming the proof has been done correctly, I am stuck in this one step which seems minor, but is holding up the rest of it. How do I prove that inclusion map $i$ is linear, for this defintion to be applicable and thus the proof to be valid?

ca.classical analysis and odes – Integrability of Fourier transform of truncated fractional power

Is the Fourier transform of the function $f$ which agrees with $1_{(-1.1)}|x|^alpha$ on $(-1,1)$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(mathbb R)$, where $alphain(1,2)$? My guess is that the answer is true. If I can show that the identity $widehat{D^alpha f}(zeta)=(2pi izeta)^alphahat{f}(zeta)$ holds in some sense and we have $D^alpha(1_{(-1,1)}|x|^alpha)=alpha!|x|1_{(-1,1)}$ in some sense. Then $hat{f}$ would have enough decay at infinity to make it in $L^1(mathbb R)?$ May be fractional derivatives and their interaction with the Fourier transform might be helpful. But I could not find any good reference (easily readable) which can make the above argument rigorous? For my purpose if one such $f$ exists for which the Fourier transform is integrable is enough?

sequences and series – Rate of decay of Fourier coefficients on heat equation solution

Let $u(x,t) = sum_{j=1}^infty u_j^0 e^{-j^2t}sin(jx)$ be the solution for the heat equation, where $u_j^0 = sqrt{(2/pi)}int_0^pi u^0(x)sin(jx)dx$ are the Fourier coefficients of the initial data $u^0$.

Suppose that $u_j^0 = C/j$.
Prove that $||frac{d}{dt}u(.,t)||leq frac{C}{t^{3/4}}$

This is a problem from “Numerical Solution of Partial Differential Equations by the Finite Element Method”, by Claes Johnson. I am having trouble to relate the decay rate of the Fourier coefficient with the boundness on the time derivative. The book also uses as an example that $||frac{d}{dt}u(.,t)||leq frac{C}{t^{1/4}}$ when $u_j^0 = C/j^2$. While I can see that $||frac{d}{dt}u(.,t)||leq frac{C}{t}$ is true because:

$frac{d}{dt}u(x,t) = sum_{j=1}^infty u_j^0 (-j^2)frac{t}{t} e^{-j^2t}sin(jx) leq sum_{j=1}^infty u_j^0 frac{C}{t}sin(jx)$

I don’t see how I can use that $u_j^0 = C/j$ to change the value from $1$ to $frac{3}{4}$.

Plotting Fourier Image of 2D function

I have function of two variables which was calculated numerically for grid {x,y}. Then I calculate numerical Fourier trasform of my function and obtain array of Fourier transform values. How can I find the array {u,v,f[u,v]}, where u and v Fourier coordinates and f[u,v] is obtained values of Fourier transform, i.e. how can I create frequency array {u,v}?

Numerical Fourier Transform of bivariate functon

I have a function $f(x,y)$ which was calculated on lattice/grid, so I have arrray

array={x,y,f(x,y)}

I would like to find Fourier transform of this function. Can I just use Fourier(array)? I have read documentation but still confuse.

Structure factor of fourier transform of an image

In this article authors show crystal and liquid phase from two dimensional crystals by calculating structure factor (Fourier transform of 2D points).

enter image description here

I have generated set of points in 2D that represent lattice points of a perfect triangular lattice and a non perfect lattice. (images below)

perfect lattice
enter image description here
not perfect lattice
enter image description here

I want plot similar results from figure 2 of the article (image below) for both cases perfect and not perfect lattice. Can I do this in mathematica?

not perfect
The plot here shows for totally random points (left), not perfect lattice (middle), and perfect lattice (right)

Thank you.

Criterion for trigonometric series to be a Fourier series for some function using Fejer Theorem

I have the following exercise:

“Prove, that a trigonometric series is a Fourier series of some function iff its Cesaro means converse uniformly”

Now, the “only if” part is Fejer’s Theorem.

I am trying to prove the “if” part.
Suppose that Cesaro means converse uniformly to some function.
I want to prove that then Fourier series for this function is the given trigonometric series.
If there is some trigonometric series $$sum_{k=1}^{infinity} u_k.$$ then the Cesaro means of its partial sums are $$s_n = sum_{k=0}^{n-1} (1-k/n)u_k.$$

Now I am trying to calculate the Fourier coefficients of f (which is the function Cesaro means universally converge to), and show that they are the same as in $$sum_{k=1}^{infinity} u_k.$$.

But I can’t do this, I only get zeroes as coefficients.
I would appreciate any help!

fast fourier transform – Butterfly diagram from cooley-tukey algorithm

I am trying to understand the logic of this algorithm so i can implement my own but i am not understanding this diagram i see appearing many times in a fair few articles on the topic, i am teaching myself so i don’t have a computer science degree to help. But i understand the general idea behind the algorithm.

But this butterfly diagram is confusing me.

enter image description here

How do you interpret this diagram from left to right mathematically speaking? It’s so confusing.. what are the arrows telling me to do in terms of math operations on the functions?

fourier analysis – Parseval-Plancherel identity involving absolute value

Let $hat{f}$ be the fourier transform of $f$.

By Parseval-Plancherel identity, for suitable $f,g$, we have
$$left|hat{f}*hat{h}right|_{L^2_{xi}}^2=left|fcdot hright|_{L^2_{x}}^2.$$

Let $f,g$ be good enough functions. I want to know if for some universal constant $C>0$,
$$left||hat{f}|*|hat{h}| right|_{L^2_{xi}}^2leq Cleft||f|cdot|h|right|_{L^2_{x}}^2.$$

Or is there a counterexample?

Thanks a lot!

fourier transform – Removing Gibbs Phenomenon

I am working with a sample of 20 points given from an unknown 1-periodic function that are plotted like this: Original sample

I am using Inverse Fast Fourier Transform (ifft) to recover the signal resampled in 1000 points at (0,1) that is plotted like this: Resampled

It is showing a Gibbs Phenomenon at the end of the signal. What can be causing this fact? As far as I know Gibbs Phenomenon occurs near a jump discontinuity…

Any idea about why is this happening and how could I solve it?