Spline Interpolation of Data of Power Growth Applied to Discrete and Continuous Riesz Means

By making use of recent results on interpolation of data of power growth by polynomial spline functions of odd degree, we obtain new relations between discrete Riesz summability (R, x, 2m-1) (i.e., of type x and odd integer order 2m-l) and the corresponding continuous Riesz summability (R, x, 2m-1), in terms of the basis chosen for the null-splines. The condition on the type-sequence (= knot-sequence) x requires that its mesh-ratio should not grow too rapidly. We relate our results to existing ones, both for cardinal knots and geometric knots. The study illustrates the important inter-relation between results in Riesz summability and those in spline interpolation.