Solution of the Cauchy problem by means of weighted approximations of exponents by polynomials (Q761627)

Let A be a positive selfadjoint operator in a Hilbert space H. It is known that the solutions \(u_ 1,u_ 2\) to the parabolic and hyperbolic Cauchy problems \(u'\!_ 1(t)=-Au_ 1(t)+f_ 0,\) \(u_ 1(0)=f_ 1,\) \(u'\!_ 2(t)=-Au_ 2(t)+f_ 0,\) \(u_ 2(0)=f_ 1,\) \(u'\!_ 2(0)=f_ 2;\) \(t>0\), are given by \(u_ i(t)=\sum_{j}G^{ij}(t,A)f_ j,\) \(i=1,2\), where \(G^{ij}(t,A)\) are exponential type functions of A. In order to by-pass the non-trivial problem of finding out all eigenfunctions of A the author gives approximation formulas \(u_ i(t)=\lim_{n\to \infty}\sum_{j}P_ n^{ij}(t,A)f_ j,\) \(i=1,2\), where \(P^{ij}\) are Lagrange interpolation polynomials of \(G^{ij}\). As a consequence he obtained results on the regularity of solutions \(u_ 1,u_ 2\).