### Approximation theory

### Daghestan Electronic Mathematical Reports, Issue 9 (2018)

# Fast computation of linear combinations of Sobolev functions generated by the Haar functions

### UDK: 519.622

### Pages: 7 - 14

### DOI: 10.31029/demr.9.2

Some properties of orthogonal with respect to Sobolev inner product functions $\chi_{1,n}(x)$, generated by Haar functions, are considered. In particular, recurrent formulas for $\chi_{1,n}(x)$ are obtained. It is developed an algorithm for calculation of the linear combinations of $N$ functions $\chi_{1,n}(x)$ using $O(\log N)$ operations.

**Keywords: **
Haar system, numerical method, Sobolev type inner product, fast algorithm.