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Click here to download Katrina Kaif XXX image gallery in jpeg format. Thanks for visiting Katrina Kaif XXX picture gallery.y_i(t),\dots,y_N(t)]^\top$ and ${{\mathbf{y}}}^\dagger$ the respective distinct localized states, and ${{\mathbf{W}}}$ the orthogonal transform matrix from orthogonal decomposition with Singular-Value Decomposition (SVD).
Note that the localization of ${{\mathbf{y}}}$ in the vector space $\mathcal{L}^2({\mathbb{R}})$ is based on the squared $L^2$-norm. This is the same for any localization function $h$ satisfying the conditions and. In the example of ${{\mathbf{y}}}(t)$ given in, the localization of the corresponding signal is that any difference between $x_i$ and its closest location is at least $2$ (with respect to the $L^2$-norm of the original signal) and any amplitude $y_i$ of the original signal is within an interval of length $2$ around its closest location.
The mapping function $\phi$ is given by $\label{eq:mc}
\phi(y_i(t)) = y_i(t) = \bar{y}_i = \frac{1}{N}(1,\dots,1)^\top \text{ for } i=1,\dots,N\text{ and }t\in[0,T_c],$ with $T_c$ the duration of the experiment (e.g. $T_c=12$).
To illustrate this method on a simple case, we consider an arbitrary Gaussian process $y_i(t)=y_i(t)$, with covariance function $C(t,s)=\sigma_{0}^2\exp[-0.5(\tau_0|t-s|-\sqrt{2\log(2)})^{2}]$, and we use as a localization function the squared $L^2$-norm. The Fourier spectrum of this process can be easily shown as follows: $\hat{C}(\omega) = \sigma_{0}^2\exp[- 0b46394aab
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