The work of Tsumura

Until the results of Shadrin (1994), there were very few results that dealt with the derivatives of the error in Hermite interpolation in other than a trivial way. By far the most outstanding of these results is ascribed to Tsumura (1941).

Though of great interest, no proof of Tsumura's result (which gives bounds of the derivatives of the error in the max norm over an interval whose endpoints are points of interpolation) has ever been published, save Tsumura's original paper.

None of those quoting the result of Tsumura whom I contacted had ever seen a proof of it. Thanks to the great efforts of Mei Kobayashi (IBM research labs, Tokyo) I was able to obtain the following copy of Tsumura's 1941 paper.

Kokai zyobibunhoteisiki ni tuite, M. Tsumura, Kansu Hoteisiki 30, pp 20-35

Tsumura's result is a simple corollary of Shadrin's recent work. Shadrin and Waldron (limited by their lack of Japanese language skills) are currently trying determine exactly the argument used in the Tsumura paper.

Last modified by Shayne Waldron on: Oct 10, 1995