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Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson, T. A. Gillespie (2001)

Studia Mathematica

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ₙ ( U ) n = 1 ∞ the sequence of (C,2) weighted discrete ergodic averages of U, that is, ₙ ( U ) = 1 / n ∑ 0 < | k | ≤ n ( 1 - | k | / ( n + 1 ) ) U k . We show that this sequence ₙ ( U ) n = 1 ∞ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

Statistical extensions of some classical Tauberian theorems in nondiscrete setting

Ferenc Móricz (2007)

Colloquium Mathematicae

Schmidt’s classical Tauberian theorem says that if a sequence ( s k : k = 0 , 1 , . . . ) of real numbers is summable (C,1) to a finite limit and slowly decreasing, then it converges to the same limit. In this paper, we prove a nondiscrete version of Schmidt’s theorem in the setting of statistical summability (C,1) of real-valued functions that are slowly decreasing on ℝ ₊. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ℝ ₊. In the proofs we make use of two nondiscrete...

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