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Paper   IPM / M / 55
School of Mathematics
  Title:   Reconstruction in a generalized model for translation invariant systems
  Author(s):  A. Daneshgar
  Status:   Published
  Journal: Fuzzy Sets and Systems
  Vol.:  83
  Year:  1996
  Pages:   51-55
  Supported by:  IPM
We consider translation invariant (TI) operators on Φ, the set of maps from an abelian group G to Ω∪{−∞}, called LG-fuzzy sets, where Ω is a complete lattice ordered group. By defining Minkowski and morphological operations on Φ and considering order preserving operators, we prove a reconstruction theorem. This theorem, which is called the Strong Reconstruction Theorem (SRT), is similar to the Convolution Theorem in the theory of linear and shift invariant systems and states that for an order preserving TI operator Y one can explicitly compute Y(A), for any A, from a specific subset of Φ called the base of Y. The introduced framework is a general model for the theory of translation invariant systems, and SRT shows the consistency of it. For the special cases when G,Ω ∈ {\BbbR,\BbbZ}, SRT implies the results of Maragos and Schafer (1985, 1987) for set-processing, function-set-processing and function-processing filters.

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