Spectral function series representation

The coefficients of the sines and cosines will be real for real data. Restoring a high-frequency band of c (unique complex) discrete spectral components to a low-frequency band of b (unique complex) spectral components will be the same (when transformed) as forming the discrete Fourier series from the high-frequency band and adding this function to the series formed from the low-frequency band. When applying the constraints in the spatial domain, the Fourier series representation will be used. 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

The intensities of the spectral lines and the depolarization coefficients are functions of the reduced matrix elements of the polarizability tensor calculated by vibronic functions. In order to estimate the possibility of observation of the pure rotational Raman spectra under consideration, one has to consider in more detail the polarizability operator. Its components belonging to the line y of representation f can be presented in the form of a power series with respect to the displacement qriri active in the Jahn-Teller effect (the other components can be neglected as not active in the pure rotational Raman spectrum under consideration) ... 

Dynamic Response Functions. - The perturbation series formula or spectral representation of the response functions can be used only in connection with theories that incorporate experimental information relating to the excited states. Semi-empirical quantum chemical methods adapted for calculations of electronic excitation energies provide the basis for attempts at direct implementation of the sum over states (SOS) approach. There are numerous variants using the PPP,50,51 CNDO(S),52-55 INDO(S)56,57 and ZINDO58 levels of approximation. Extensive lists of publications will be found, for example, in references 5 and 34. The method has been much used in surveying the first hyperpolarizabilities of prospective optoelectronically applicable molecules, but is not a realistic starting point for quantitative calculation in un-parametrized calculations. 

The spectral methods are based on using a representation of the solution /(, z) throughout the domain L2 via a polynomial trial function expansion (Eq. 12.406). A nodal basis expansion in terms of the Lagrange basis polynomials is obtained by letting (z) = idiz)and j it = f ( ). Thus, the series expansion is given by Eq. (12.408), or alternatively by Eq. (12.441) in terms of global indices. The resulting residual functions are thus in terms of the solution function values =/> at the... 

The most widely used is the spectral representation method, by which a stochastic process is regarded as the sum of a series of harmonic functions with random phase, i.e.,... 

Since the KL expansion is obtained as a linear sum of Gaussian random variables, the series can be used to represent only Gaussian random processes. For spectral representation of non-Gaussian random processes, one uses the more general form known as the polynomial chaos expansion. Polynomial chaos expansion is a spectral representation of the random process in terms of orthonormal basis functions and deterministic coefficients. Based on the Cameron and Martin theorem (Cameron and Martin 1947), it can be shown that a zero mean, second order random process can be represented as... 

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