Optimal recovery of mappings based on linear information with the help of $T$-splines in Banach spaces

This work is dedicated to solving problems of optimal recovery of operator $A$ (not necessarily linear), defined on a subset $\mathfrak{M}$ of a Banach space $H$ using information about elements of the $\mathfrak{M}$, given by a linear bounded operator $T\colon H\to Y$ where $Y$ is some Banach space. We show that under certain condition the optimal method of recovery is given by abstract interpolation splines in $H$ generated by $T$ ($T$-interpolating splines).