Transformation of the second order modulus by positive linear operators
Abstract We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.
AN INVERSION FORMULA FOR LAPLACE TRANSFORMS AND SEMI- GROUPS OF LINEAR OPERATORS'
This paper is primarily concerned with the problem of determining necessary and sufficient conditions that a closed linear operator be the infinitesimal generator of a semi-group of linear bounded transformations. The first results in this direction were published independently by E. Hille [3] and K. Yosida [8] in 1948. They found necessary and sufficient conditions on the resolvent R(X; U) of an operator U in order that U generate a semi-group T(s) such that 1f T(s) 11 f exp (ws) for some real w. Hille has recently given another proof of this theorem [4] based entirely on the Post-Widder inversion formula for Laplace transforms. Yosida, in his proof, introduced the approximating semi-groups
UNIVERSAL LOCALLY COMPACT SCATTERED SPACES
If δ is an ordinal, we denote by C(δ) the class of all cardinal sequences of length δ of locally compact scattered (in short: LCS) spaces. If λ is an infinite cardinal, we write Cλ(δ) = {s ∈ C(δ) : s(0) = λ = min[s(ζ) : ζ < δ]}. An LCS space X is called Cλ(δ)-universal if SEQ(X) ∈ Cλ(δ), and for each sequence s ∈ Cλ(δ) there is an open subspace Y of X with SEQ(Y ) = s. We show that • there is a Cω(ω1)-universal LCS space, • under CH there is a Cω(δ)-universal LCS space for every ordinal δ < ω2, • under GCH for every infinite cardinal λ and every ordinal δ < ω2, there is a Cλ(δ)-universal LCS space, • there may exist a Cω(ω2)-universal LCS space. As a consequence, we obtain that it is consistent that 2 = ω2 and Cω(ω2) is large as possible, i.e. Cω(ω2) = {s ∈ 2{ω, ω1, ω2} : s(0) = ω}. 2000 Mathematics Subject Classification. 54A25, 06E05, 54G12, 03E35.