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TRANSLATION, MODULATION AND DILATION SYSTEMS IN SET-VALUED SIGNAL PROCESSING
(Vasyl Stefanyk Precarpathian Natl Univ, 2018) Levent, H.; Yilmaz, Y.
In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the pth power of their norm is integrable. In general, this space is denoted by L-p (R, Omega(C)) for 1 <= p < infinity and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on L-2 (R, Omega(C)) and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on L-2 (R, Omega(C)) is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space L-2 (R, Omega(C)) is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space L-2 (R, Omega(C)).