The Life and Work of D.H. Hyers, 1913-1997

The following is a sketch of the life and work of Donald Holmes Hyers, Professor Emeritus from the University of Southern California. The theorem put forth by Hyers in 1941 concerning linear functional equations has gained a great deal of interest over the past two decades. Hundreds of articles have been written citing his works, many of which have furthered the theorem. This paper contains a brief description of Hyers’ theorem, a biographical essay and an extensive bibliography of Hyers’ work and works citing the Hyers theorem or the D.H. Hyers–S.M. Ulam–Th.M. Rassias theorem or related subjects of almost the last three decades. The author of this paper is the grandson of D.H. Hyers

Approximate testing with error relative to input size

We formalize the notion and initiate the investigation of approximate testing for arbitrary forms of the error term. Until now only the case of absolute error had been addressed ignoring the fact that often only the most significant figures of a numerical calculation are valid. This work considers approximation errors whose magnitude grows with the size of the input to the program. We demonstrate the viability of this new concept by addressing the basic and benchmark problem of self–testing for the class of linear and polynomial functions. We obtain stronger versions of results of Ergün, Ravi Kumar, and Rubinfeld [11] by exploiting elegant techniques from Hyers–Ulam stability theory