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
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.