Translation, modulation and dilation systems in set-valued signal processing

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 $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ 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 $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ 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 $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$.