On wavelet decomposition of spaces of first order splines

We consider approximate relations in the form of a system of linear algebraic equations that yield B _φ -splines. We construct Lagrange type splines of the first order and give examples of polynomial, trigonometric, hyperbolic, and exponential B _φ -splines. We also construct a system of linear functionals biorthogonal to the B _φ -splines and resolve an interpolation problem generated by this system. For refined nonuniform grids we establish an embedding of spaces of B _φ -splines. The decomposition and reconstruction formulas are obtained. Bibliography: 20 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 47–60.