Smooth approximations in $C[0, 1]$

The first orthonormal basis in the space of continuous functions was constructed by Haar in 1909. In 1910, Faber integrated the Haar system and obtained the first basis of continuous functions in the space of continuous functions. Schauder rediscovered this system in 1927. All functions of Faber – Shauder are piecewise linear, and partial sums are inscribed polygons. There was many attempts to build smooth analogues of the Faber – Schauder basis. In 1965, K. M. Shaidukov succeeded. The functions he constructed were smooth, but consisted of parabolic arcs. Shaidukov proved the uniform convergence of the obtained expansions, but failed to obtain deviation estimates. Another analogue of the Faber – Schauder system was proposed by T. U. Aubakirov and N. A. Bokaev in 2007. They built a class of functions that form a basis in the space of continuous functions, obtained estimates of the deviation of partial sums from the approximated function. The functions constructed were piecewise linear, as in the Faber – Schauder system. We construct smooth analogues of the Faber – Schauder system and obtain estimates of the deviation of partial sums from the approximate function. Those are systems of compressions and shifts of a single function, which we call the binary basic spline. The binary basic spline is an integral of the $n$-th order of the Walsh function $W_ {2^n-1}$. Thus, we were able to construct analogues of the Faber – Schauder system with a large degree of smoothness and obtain deviations in terms of module of continuity.