Parsevals equation

Suppose we are given a complex function / (x) defined on an interval [a, 6] and an orthonormal set of functions pi (.x) where x 6 R is real. We may then expand /(.x) in terms of the functions ipi (.c) according to 

Basis sets and the canonical ensemble partition function 

A special case of Eq. (B.65) is obtained if the function / itself is normalized to 1 that is, 

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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. 

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

Definition 51 An orthonormal and complete set of elements e, in a Hilbert space H is called an orthonormal basis of Hilbert space. The numbers x, Cj) are called the Fourier coefficients of x, the expression x = x,ei)ei is called the Fourier expansion of x, and the equation (A.44) is called Parseval s equation. ... 

As time increases from —oo to 0, the half width of the wave packet y(x, t) continuously decreases and the maximum amplitude continuously increases. At t = 0 the half width attains its lowest value of y/lja and the maximum amplitude attains its highest value of 1 / and both values are in agreement with the wave packet in equation (1.20). As time increases from 0 to oo, the half width continuously increases and the maximum amplitude continuously decreases. Thus, as increases, the wave packet y(x, z) remains gaussian in shape, but broadens and flattens out in such a way that the area under the square y(x, 01 of the wave packet remains constant over time at a value of (2A/7ra), in agreement with Parseval s theorem (1.18). 

Non-uniform FT (nuFT) emplo3dng an equation identical to (35), but with nonquadratic matrix A (or with full m x m matrix A and zeros at non-sampled points of w-long vector s). The obtained solution features minimum Z2-norm (power), which can be easily proved, considering that FT is an unitary operation and thus Zy-norms of signal and spectrum are equal (Parseval s theorem). Although, the solution is not the optimal one, the processing is fast and was successfully... 

Corollary 5.6 Parseval s Theorem. The variance of a signal can be computed from its spectral density using the following equation... 

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