A property of the finite-difference second-derivative operator

We consider the finite-difference eigenvalue problem u _xx ^– + u=0, u₀= u_n+1=0 on a nonuniform grid =x_ii=0,1,...,n+1, x₀=0, x_n+1=1. In connection with the issue of existence of exact-spectrum schemes for second-derivative operators, we examine the extremal properties of functions f_n(v, h)=₁ ^v(h)+ ...+_{n v}(h), v R. We prove that the maximum of f_n(–1, h) is attained only on a uniform grid. We establish a necessary condition for given numbers 0 <₁ <... < _n to be the eigenvalues of the above problem for at least one grid .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 3–8, 1987.