On the almost chebyshevian approximation to certain operators of the hereditary theory of elasticity

Abstract

Resolvent hereditary operator generated by an integral, Volterra type operator with the Abel /1/ or Rzhanitsyn /2/ kernel is approximated on an arbitrary, finite time interval, by a polynomial in fractional powers of a variable with an exponential co-factor, using the method developed in /3,4/. The approximation obtained from all polynomials of the given type offers, firstly, a smallest error in the defining equation, and secondly, it approaches asymptotically with increasing order the Chebyshev polynomials of the best uniform approximation to the function on the segment. The estimation of the approximation obtained shows that the error decreases with increasing order of approximation at least as rapidly as the geometrical progression.