In 1956 V.A. Il'in proved that the eigenfunction expansion of a piecewise smooth function associated with the Laplace operator in an arbitrary two-dimensional domain converges uniformly on every compact supset where the function is smooth. Recently (for last 10 years) this problem for elliptic operators of second order was studied in different works of authors M.Taylor, M.Pinsky, L.Brandolini, L.Colzani, Sh.Alimov and others. The main purpose of the lecture is to show that the same results are valid for elliptic operators of an arbitrary order m.