The integral values of some successive equidistant subintervals are known in practical areas

whereas the usual function values at the knots are not given in interpolation problems. We propose a multilevel integro cubic spline quasi-interpolation for function approximation from given integral values over successive subintervals and multilevel spline quasi-interpolation. We used the linear combination of the given integral values to approximate function values at knots. The multilevel cubic spline quasi-interpolation operator was defined with the classical cubic spline quasi-interpolation and its corresponding error function. Finally

we obtained its polynomial reproducing property and error estimate. The proposed method

together with the existing integro cubic spline quasi-interpolation

was tested by two infinitely differentiable functions. Numerical experiments showed that the proposed method possessed better approximation behaviors and numerical convergence orders compared with the integro cubic spline quasi-interpolation. Multilevel integro cubic spline quasi-interpolation can successfully approximate the original function and its first and second-order derivative functions over the global interval. This process has good approximation behavior and numerical convergence compared with the existing integro spline quasi-interpolation. Moreover

the proposed method of function reconstruction from the integral values of successive subintervals is universally applicable.