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Additional info for A new approach to linear filtering and prediction problems
5-1) whose coefficients satisfy Hypotheses 11-2 and 11-3 will be said to be in Arnold's form. The fewer blocks there are in the partition of A0, or, in other words, the fewer elementary divisors A0 possesses, the sparser the matrices Ar will be. 1 may help the reader visualize the situation. It illustrates the case that A. has three elementary divisors. The heavy lines indicate the last rows or first columns of the blocks of the Ar. Only they may contain nonzero entries. 4-3). The exponent a is, as yet, undetermined.
1-3) The expansion is uniform in every compact subdomain of D. 1-3) may be indefinitely differentiated termwise with respect to both variables. The preceding theorem is remarkably general: The functions ar may grow arbitrarily fast with r, and the constants a, /3, eo are arbitrary as long as 0 < /3 a < 277. Even that condition can be eliminated by defining E as a sector of a Riemann surface with a branch point ate = 0. It is obvious that e = 0 cannot be an interior point of E, except when the series converges.
Denote by J(x) the Jordan form of A0(x). If Hypothesis II-1 is satisfied, then there exists a series E OprEr such that the II. 3-19) with the following properties. (i) All Pr and Br are holomorphic in D; (ii) det Po(x) 0 for all x E D; (iii) All Br are block diagonal. The orders of these blocks are the multiplicities of the eigenvalues of A0 in D, and each block has only one distinct eigenvalue. (iv) Bo = J. The simplest case is the one in which p = n. Then the problem has been formally reduced to a diagonal system, which can be formally solved by quadratures and straightforward manipulations with series.