On End Conditions for Integro Quadratic Spline Interpolation in the Mean

We investigate a problem of function reconstruction, where instead of function values at certain points of the interval, integrally averaged values over intervals are known. Yu. N. Subbotin proposed to call such problems interpolation in the mean. Using a relationship between integro quadratic splines that solve this problem and interpolating cubic splines, we discuss the question of choosing end conditions in the case when no additional information is available at the endpoints of the interval that could be used as end conditions. For the construction of the spline interpolating in the mean via B-splines, we propose formulas that explicitly give the expansion coefficients at the endpoints of the interval which do not change the maximally possible third order of approximation.