Dirac delta function spectral density

Another approach to approximating the scaling functions is to construct some model spectral densities. In turn, Eqs. (14)—(19) yield the approximate Fe and Fx. The choice of the Dirac delta function for T yields trivial relations... 

On the other hand, the undamped autocorrelation function (17) we have obtained within the standard approach avoiding the adiabatic approximation must lead after Fourier transform to spectral densities involving very puzzling Dirac delta peaks given by... 

How does one extract eigenpairs from Chebyshev vectors One possibility is to use the spectral method. The commonly used version of the spectral method is based on the time-energy conjugacy and extracts energy domain properties from those in the time domain.145,146 In particular, the energy wave function, obtained by applying the spectral density, or Dirac delta filter operator (8(E — H)), onto an arbitrary initial wave function ( (f)(0)))1 ... 

The approach to the modeling consists of assuming the bilateral power-spectral density function as the sununatitHi of Dirac s delta functions with unknown amplitudes, that is,... 

As also showed in Eq. (5.257), the transition probability or the conditioned probability density p x, t, x, t can be seen as a special probability density with the initial condition given by the delta Dirac function. In such conditions, for the conditioned probability density one can obtain the coefficients of the spectral expansion in Eq. (5.289) ... 

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