The Translation Group of Lp Functions, and Moduli of Continuity Lp.
Let let a function of class and let The h-translation of, that is the function
belongs to the class; moreover, if, there holds
as
Therefore, since translations are in fact linear isometries, also
as
uniformly on .
In other words, the map defines a strongly continuous group of linear isometries of . In the case the above property does not hold in general: actually, it exactly reduces to the uniform continuity, and defines the uniform continuous functions. This leads to the following definition, that generalizes the notion of a modulus of continuity of the uniformly continuous functions: a modulus of continuity for a measurable function is a modulus of continuity such that
This way, moduli of continuity also give a quantitative account of the continuity property shared by all functions.
Read more about this topic: Modulus Of Continuity
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