Orthogonal expansion

The basis behind separation of variables is the orthogonal expansion technique. The method of separation of variables produces a set of auxiliary differential equations. One of these auxiliary problems is called the eigenvalue problem with its eigenfunction solutions. 

The expansion coefficients r were obtained by orthogonal expansion their values are shown in Table 1. 

When the region of expansion is properly chosen, the error e x) oscillates on both sides of the abscissa. Thus, choosing an orthogonal polynomial P x) is equivalent to demanding that the error of the approximation be zero at a finite set of points. This is in contrast to the Taylor series for which the error is zero only at one point. In this sense the orthogonal expansion is an interpolating approximation. 

Bigeleisen and Ishida (7) obtained an orthogonal expansion of the logarithmic function. 

R.E. Clapp, A complete orthogonal expansion for the nuclear three-body problem. Ann. Phys., 13 187-236,1961. 

Wo now recall that Fourier expansions (F.18) are orthogonal expansions (sec, e.g., Ref. 242). It follows that each sununand on the second line of Eq. (F.20) has to be equal to its counterpart on the third line so that... 

This problem can be overcome by the use of orthogonal expansion functions (the subject of Sec. III.B). Another possible numerical fix is to use more sampling points than functions. The overdetermination of the inversion is overcome by a least squares procedure (18). 

To find an approximate solution of the kinetic equation, an orthogonal expansion of the velocity distribution with respect to the direction v/v of the velocity v is commonly used in the treatment of the kinetic equation. Depending on the... 

The Wiener kernels depend on the GWN input power level P (because they correspond to an orthogonal expansion), whereas the Volterra kernels are independent of any input characteristics. This situation can be likened to the coefficients of an orthogonal expansion of an analytic function being dependent on the interval of expansion. It is therefore imperative that Wiener kernel estimates be reported in the literature with reference to the GWN input power level that they were estimated with. When a complete set of Wiener kernels is obtained, then the complete set of Volterra kernels can be evaluated. Approximations of Volterra kernels can be obtained from Wiener kernels of the same order estimated with various input power levels. Complete Wiener or Volterra models can predict the system output to any given input. 

To reduce the requirements of long experimental data records and improve the kernel estimation accuracy, least-squares methods also can be used to solve the classical linear inverse problem described earlier in Equation 13.6, where the parameter vector 9 includes all discrete kernel values of the finite Volterra model of Equation 13.17, which is Knear in these unknown parameters (i.e., kernel values). Least-squares methods also can be used in connection with orthogonal expansions of the kernels to reduce the number of unknown parameters, as outlined below. Note that solution of this inverse problem via OLS requires inversion of a large square matrix with dimensions [(M -I- f -I- 1) /((M -F 1) / )], where M is... 

The most general (functional) approach to evaluate an operation is to follow a domain-partitioning paradigm. This involves computing orthogonal expansions for arguments and result and than convert the result to the desired decomposition type, i.e. use function succ to obtain graph successor functions. 

Table 1 shows examples with different runtime and space requirements. As expected, graph types differ significantly for their word-level and bit-level behaviour. While MTBDD construction at the word-level is often very fast, BMDs and p BMDs tend to be more compact at reasonable runtimes. The situation is often reversed at the bit-level, where orthogonal expansions are advantageous. 

Sakhabetdinov M. A. Elaboration and Investigation of an Orthogonal Expansion Method for the Analysis of Experimental Data about Thermophysical Properties of Substances Using Computers. Author s Abstract of Candidate Thesis. MEI, Moscow, 1977. 

In Sects. 14.3,15, and 16 we shall benefit from the exercise we got in handling overlap effects. In the rest of the book we shall deal with usual orthogonal expansions. 

---