Approximations by Max-Product Sampling Operators

In this chapter we introduce and study the max-product sampling operators, which have applications to signal theory. Due to the fact that for bounded functions with positive values, the max-product sampling operators attached to them have nice properties, all the approximation results in this chapter are stated and proved under this restriction. But as it was already mentioned in Subsection 1.1.3 , Property C, this restriction can easily be dropped by considering the construction used for the max-product Bernstein operator in Theorem 2.9.1 . More precisely, if 𝒮 W , φ ( M ) $$\mathcal{S}_{W,\varphi }^{(M)}$$ is any max-product sampling operator defined in this chapter and f : ℝ → ℝ $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is bounded and of variable sign, then it is easy to see that the new operators P W , φ ( M ) ( f ) ( x ) = 𝒮 W , φ ( M ) ( f − a ) ( x ) + a $$P_{W,\varphi }^{(M)}(f)(x) = \mathcal{S}_{W,\varphi }^{(M)}(f - a)(x) + a$$ , where a