Approximation of the process impulse response

A sequence of real functions h t),l2 t). is said to form an orthonormal set over the interval (0, oo) if they have the property that 

A set of orthonormal functions li(t) is called complete if there exists no function f(t) with /q°° f t) dt oo, except the identically zero function, such that 

The Laguerre functions (Lee, 1960) are an example of a set of complete orthonormal functions that satisfy the properties defined by Equations (2.1)-(2.3). The set of Laguerre functions is defined as, for any p 0 

The parameter p is called the time scaling factor for the Laguerre functions. This parameter plays an important role in their practical application and will be discussed in detail in Section 2.3. (Note The set of Laguerre functions presented in Equations (2.4) differs by a factor of —1 for even values of i when compared with the set of Laguerre functions presented by Lee (1960). However, this does not affect the orthonormal properties of these functions.) 

With respect to a set of functions li[t) that is orthonormal amd complete over the interval (0,00), it is known that an arbitrary function h t) has a formal expansion analogous to a Fourier expansion (Wyhe, 1960). Such an expansion has been widely used in numerical analysis for the approximation of functions in differential and integral equations. The idea behind using Laguerre functions to represent a linear, time invariant process is to take h t) to be the unit impulse response of the process, where h t) can be written as 

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