Parseval

Using Parseval s theorem, from equation (9.126) the //2-optimal eontrol problem ean be expressed in the frequeney domain as... 

Since the function A k) is the Fourier transform of (x, t), the two functions obey Parseval s theorem as given by equation (B.28) in Appendix B... 

Show that the square pulse A k) in equation (1.21) and the corresponding function W(x, t) obey Parseval s theorem. 

Parseval s theorem is also known as the completeness relation and may be used to verify that the set of functions e " for —oo oo are complete, as (fiscussed in... 

Applying Parseval s theorem for Fourier series to the scries 22.3) and (22.4) we find that the series... 

The integration can be performed in reciprocal space, like the reciprocal-space evaluation of the electrostatic potential discussed in chapter 8. According to Parseval s rule (discussed in chapter 5),... 

Of a special interest here is a charge in aperiodic motion, as in a collisional encounter. In that case, the theory of Fourier transforms is used to describe the continuous spectra that result. Specifically, starting from Eq. 2.60 and making use of Parseval s theorem, Eq. 2.52, the total energy radiated in the aperiodic event per unit solid angle and per unit frequency interval is obtained as... 

The total power P in the excitation is obtained from Parseval s theorem ... 

The rest of the optimization is identical to the procedure described for the white power spectrum. The total excitation power in a (-1,1) binary sequence is, according to Parseval s theorem [Eq. (66)] always constant. 

It is often the case that the Franck-Condon factors contained in dq(ij) vary sufficiently slowly over the range of E encompassed by the dump pulse to be regarded as constant [191], (This assumption, called the slowly varying continuum approximation is discussed in detail in Chapter 10). Under these circumstances we can use the following generalized Parseval s equality to show that P(q) is independent of the coherence properties of the dump pulse. Specifically we have... 

The conventional Parseval equality da>( e co) 2) = dt(e (t)s(t)) is the special, case of o>i = co2.] Since the right-hand side is independent of the phase of c(f), V then the frequency-integrated correlation function is independent of the degree of jcoherence of the dump pulse. 

Parseval s theorem demonstrates that the total power is the same whether it is computed in the time domain or the frequency domain. It is often of interest to determine the power in some small interval (f+ df), as a function of the frequency. This is called the spectrum of the signal h(t). 

To obtain Parseval s theorem for the function /(x) in equation (B.17), we first take the complex conjugate of /(x)... 

Finally, integration over the variable k yields Parseval s theorem for the Fourier integral,... 

Boissel P, Marty P, Klotz A, de Parseval P, Chaudret B, Serra G (1995) Chem Phys Lett 242 157... 

Klotz A, Marty P, Boissel P, de Caro D, Serra G, Mascetti J, de Parseval P, Derouault J, Daudey JP, Chaudret B (1996) Planet Space Sci 44 957... 

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