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In these notes we lay out the basic theory of the Fourier transform, which is of course the most fundamental tool in harmonic analysis and also of major importance in related fields (functional analysis, complex analysis, PDE, number theory, additive combinatorics, representation theory, signal processing, etc.). The Fourier transform, in conjunction with the Fourier inversion formula, allows one to take essentially arbitrary (complex-valued) functions on a group $latex {G}&fg=000000$ (or more generally, a space $latex {X}&fg=000000$ that $latex {G}&fg=000000$ acts on, e.g. a homogeneous space $latex {G/H}&fg=000000$), and decompose them as a (discrete or continuous) superposition of much more symmetric functions on the domain, such as characters $latex {\chi: G \rightarrow S^1}&fg=000000$; the precise superposition is given by Fourier coefficients $latex {\hat f(\xi)}&fg=000000$, which take values in some dual object such as the Pontryagin dual $latex {\hat G}&fg=000000$ of $latex {G}&fg=000000$. Characters behave in a very simple manner…

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Fractional Brownian motion


Given a complete probability space  \left(\Omega, \mathcal{F}, \mathbb{P} \right).

A fractional Brownian motion (fBm)  \left( B_t^H, t \geq 0\right)  of Hurts parameter  H \in (0,1) is the continuous centered Gaussian process with covariance function

R^{H}(t,s):=\mathbf{E}\left[ B_t^{H} B_s^{H} \right] =\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}). (1)

 The parameter H characterizes all the important properties of the process.

Note that  E\left[(B_t^H)^2 \right]=|t|^{2H} and then \mathbf{E}\left[ (B_t^H- B_s^H)^2\right]= |t-s|^{2H}.  As \left( B^H(t), t \geq 0 \right) is Gaussian so he admits a version with  continuous trajectories or continuous modification, according to the Kolmogorov theorem.

For H=1, we set B_t^H= B_t^1= t \xi where \xi is standard normal random variable.

The  parameter  H controls the regularity of the trajectories, which are Hölder continuous of order H- \varepsilon , for any \varepsilon >0. more precisely,

For all \varepsilon > 0 and \alpha > 0, there exists a nonnegative random variable  X_{\varepsilon, \alpha} such that E\left[ |X_{\varepsilon, \alpha}|^p\right] < \infty  for all   p \geq 1, and

|B_t^H -B_s^H| \leq X_{\varepsilon, \alpha} |t-s|^{H - \varepsilon}

This is simply the  modulus of continuity for the trajectories of a  fBm \left( B_t^H, t \geq 0 \right) .

If H=\frac{1}{2}, the covariance  R_{\frac{1}{2}}(t,s)= \min(t,s)  and the process \left(B_t^H , t \geq 0\right) is a standard brownian motion.  in this case, the increments of this process  in disjoint intervals are independents.


An \mathbb{R}^d-valued random  process X= (X_t, t \in \mathbb{R}) is  bself-similar  if  for any  a >0 there exists    b > 0  such that

\displaystyle\left(X_{a t}, t \in \mathbb{R} \right)  and  \displaystyle\left( a^bX_{ t}, t \in \mathbb{R} \right)  have the same distribution.

that means for, for every choise t_0, \cdots , t_n in  \mathbb{R}  we have,

                                                                                               \displaystyle{\mathbb{P}\left( X_{a t_0}  \leq  x_0, \cdots , X_{a t_n}  \leq  x_n\right) =   \mathbb{P \left( bX_{ t_0}  \leq  x_0, \cdots , b X_{t_n}  \leq  x_n\right)}

for every x_0, \cdots , x_n in  \mathbb{R}.

Since the covariance function of the fBm is homogenous of order 2 H, we  deduce that the process \left( B_t^{H}, t \geq 0 \right) is self-similar (Put b=a^{-H}).

Note that

\mathbf{E}\left[(B_t^{H}  -B_s^{H} )(B_u^{H} -B_v^{H} )\right]= \frac{1}{2}\left[|s-u|^{2 H} + |t -v|^{2 H} - |t -u|^{2 H} - |s -v|^{2 H}\right]   (2)

and it follow that the process  \left( B_t^{H} , t \geq 0 \right) has stationary increment,  However it  is not stationary itself.

Let H \in ]0, \frac{1}{2}[ \cup ]\frac{1}{2}, 1[ and t_1 < t_2 < t_3 < t_4, it follow from  (2)  that

\mathbf{E}\left[(B_{t_4}^{H} - B_{t_3}^{H} )(B_{t_2}^{H}  - B_{t_1}^{H} ) \right]= H(2 H -1) \int_{t_1}^{t_2} \, \int_{t_3}^{t_4} (u-v)^{2 H-2} \, du \, dv.