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
On one-parameter semi-groups of linear transformations
for 0 O. We shall show that the first hypothesis for this theorem implies the second. We shall also show by means of an example that weak measurability is not sufficient to imply the boundedness of IIT(t)II in any interval [5, 1/a]. We being by proving the following lemma.