Korovkin's theorem for locally compact Abelian groups

Joseph F. Sussich

In 1953 P.P. Korovkin proved that {1,cos,sin} is a test set for each sequence (Tn) of positive linear operators on the space C of continuous real 2 π-periodic functions, in the sense that lim Tnf = f uniformly for f = 1 , cos and sin implies that lim n->∞ Tn f = f uniformly for each f ∈ C . This thesis deals with versions of Korovkin's result on LP(G), p ∈ [1,∞) , and Cu(G) (the complex vector spaces of pth-power integrable functions and of bounded uniformly continuous functions on G , respectively) where G is a locally compact abelian Hausdorff group. In Chapter One we refer to some known Korovkin results, mainly in the setting of vector lattices and using the concept of affineness, and relate these to the LP(G) and Cu(G) cases. The second chapter examines the use of sets of continuous characters as tests sets. We find that in general even the entire dual of G need not be a test set and that a restriction on the supports of the operators is necessary; though when G is compact any set of continuous characters that separates the points of G and contains the identity is a test set. From a suitable family of functions on G it is possible to define a useful modulus of continuity and to obtain quantitative Korovkin results in terms of it; this is the basis of Chapter Three. We give a general construction of such a family of functions and show that when G is compact and connected each function can be constructed from a finite set of continuous characters, so that we obtain a quantitative version of the result of Chapter Two specified in the preceding paragraph. Finally, in the appendix we consider saturation in LP(G), generalising the usual definition by introducing the notion of trivial class (those functions that are eventually perfectly approximated). We also extend a result, due to Bemd Dreseler and Walter Schempp, describing the saturation class for certain types of convolution operators on L2(G).

Korovkin's theorem for locally compact Abelian groups

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