Histospline functions in reconstructing functions using integral values

Background: Reconstructing continuous functions from their integral values over specific intervals is a significant problem in applied and computational mathematics, especially when direct node values are unavailable. Several researchers have conducted studies in this area, proposing various methods to reconstruct functions, such as traditional approaches, algebraic techniques and others. Purpose: The objective of this paper is to study the histospline method in Sobolev spaces to reconstruct continuous functions given by their interval integral values and to obtain explicit representation formulas for the coefficients. Method: The histospline method is investigated in the Sobolev spaces to reconstruct functions given by their interval integral values. It is shown that when the derivative is applied to the natural spline, the degree of the histospline function decreases by one. Results: As a result, a piecewise constant function is obtained in the Sobolev space of functions with square-integrable first-order derivatives on the unit interval, while a quadratic function is formed in the Sobolev space of functions with square-integrable second-order derivatives on the unit interval. Explicit formulas for the coefficients of the histopolating spline are derived, and numerical experiments are presented. Implication: The constructed histosplines ensure an exact match for the area of the respective histogram bars, providing an efficient framework for function reconstruction in low-order Sobolev spaces.