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On the asymptotic behaviour of \(\varphi\)-means. (English) Zbl 0833.26004

The author offers a generalization of a result by R. P. Boas and J. L. Brenner [J. Math. Anal. Appl. 123, 262-264 (1987; Zbl 0614.26001)], which states that, under certain conditions, the limit of \(f^{-1}(\sum p_k f(x_k + t)) - t\) as \(t\) goes to \(+ \infty\) is \(\sum p_k x_k\) \((\sum p_k = 1)\), to (Stieltjes) integral means on masses with compact support.

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A06 One-variable calculus
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A48 Monotonic functions, generalizations
28A25 Integration with respect to measures and other set functions

Citations:

Zbl 0614.26001
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