The article under review refers to the papers [\textit{L.~S.~Maergojz}, Sib. Math. J. 41, No. 6, 1126-1136 (2000; Zbl 0970.32011) and Dokl. Math. 56, No. 2, 674-678 (1997; Zbl 0973.32002)] wherein the problem of optimal extrapolation from a finite set is studied in the class of entire functions with finite spectrum. The aim of the present article is to study this problem in the case of analytic continuation from a finite set with inaccurate data. To estimate the error of a linear functional, an approach is employed suggested by \textit{K.~Miller} in [SIAM J. Math. Anal. 1, 52-74 (1970; Zbl 0214.14804)] based on using the least squares method for ill-posed problems with a prescribed bound. As a result, the authors obtain constructive formulas for calculating the optimal error of the optimal linear algorithm for extrapolation from a set \(U\) to a point \(z_0\) in the class of functions \[ V = \{f\in H(D): \|f\|\leq r\}, \quad r > 0, \] where \(H\) is a Hilbert space with reproducing kernel. Moreover, the asymptotic behavior of the optimal error is investigated in the case when the errors of estimating the initial data tend to zero.