Approximation of piecewise smooth functions by a small binary basis from eigenfunctions of the two Sturm–Liouville problems under mutually symmetric boundary conditions

A method for construction of specific basis functions is formulated. This method is based on eigenfunctions of the two general Sturm–Liouville problems under two different mutually symmetric versions of boundary conditions. The expansion of smooth and piecewise smooth functions leads to rapidly convergent series. This result is the basis for approximation of the above-mentioned functions by means of a small number of terms.