By Dr. R. Saeks (auth.)
If one takes the intuitive standpoint procedure is a black field whose inputs and outputs are time capabilities or time sequence it truly is typical to undertake an operator theoretic method of the research of such structures. the following the black field is modeled through an operator which maps an enter time functionality into an output time functionality. Such an process yields a unification of the continual (time functionality) and discrete (time sequence) theories and concurrently permits one to formulate a unmarried thought that's legitimate for time-variable disbursed and nonlinear structures. unusually, notwithstanding, the nice capability for such an strategy has only in the near past been discovered. Early makes an attempt to use classical operator conception mostly having failed whilst optimum controllers proved to be non-causal, suggestions platforms volatile or coupling networks non-lossless. in addition, makes an attempt to bypass those problems by means of including causality or balance constraints to the issues failed while it used to be discovered that those time dependent suggestions have been undefined and; in truth, undefinable; within the Hilbert and Banach areas of classical operator theory.
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Additional info for Resolution Space, Operators and Systems
In fact, it is possible to exhibit bounded (and even compact) operators for which the required integrals of strict triangular truncation fail to exist even in the strong topology hence even "nice" operators may fail to have an additive decomposition (DE-l. GO-3). Finally, one special class of convolution operators wherein the above C, M and A are assured to exist in the uniform sense is the case where T is an Ll sequence. hence the 46 required decomposition does exist given reasonable assumptions on the operator T (SA-7).
Of course, similar equalities hold with the integrals of triangular truncation replaced by their various equivalents and/or the entire equality replaced with strongly convergent integrals. (k,j)u(j) j .. (k,j) = 0 for k ~ < j. (k,k)u(k) where the existence of the strongly convergent integral is assured by the fact that the given representation is well defined for each u in L 2 (Z,R,m). (k,k) = 0; hence, our contention is verified. We note that the above argument fails for strictly causal operators since the diagonal integral may fail to converge uniformly (though the representation given for the diagonal integral is valid when it converges uniformly).
The concept of strict causality is, physically, somewhat ambiguous and therefore can be given a number of distinct mathematical interpretat ions as we have already seen via our study of the relationships between strong strict causality and strict caUSality. Clearly, we could define weak strict causality by taking the limit for the integrals of triangular truncation in the weak operator topology which would yield a class of operators larger than the strongly strictly causal operator (since strong convergence implies weak convergence).
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