On the Best Ulam Constant of the Linear Differential Operator with Constant Coefficients

The linear differential operator with constant coefficients D(y)=y(n)+a1y(n−1)+…+any,y∈Cn(R,X) acting in a Banach space X is Ulam stable if and only its characteristic equation has no roots on the imaginary axis. We prove that of D distinct rk satisfying Rerk>0,1≤k≤n, then best KD=1|V|∫0∞|∑k=1n(−1)kVke−rkx|dx, where V=V(r1,r2,…,rn) Vk=V(r1,…,rk−1,rk+1,…,rn),1≤k≤n, are Vandermonde determinants.