Functions with Bounded nth Differences

Abstract

Let S be a commutative semigroup with a zero element and Y a real Banach space. For functions f: S → Y, we define the difference operator △ _h f (x)by △ _h f (x)= f (x + h) - f (x). Similarly, we define\(\Delta _{{h_1}{h_2}}^2f\left( x \right) = {\Delta _{{h_2}}}\left( {{\Delta _{{h_1}}}f\left( x \right)} \right)and\Delta _{{h_1} \cdots {h_{n + 1}}}^{n + 1}f\left( x \right) = {\Delta _{{h_{n + 1}}}}\left( {\Delta _{{h_1} \cdots {h_n}}^nf\left( x \right)} \right),n = 1,2,....\)Note that the nth difference is symmetric in the increments h₁,…, h _n . When all the increments are equal to h we write\(\Delta _h^nf\left( x \right)\)instead of\(\Delta _{h \cdots h}^nf\left( x \right)\).