### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 8 (2017)

# The inversion of the Laplace transform by means of generalized special series of Laguerre polynomials

### UDK: 517.538

### Pages: 7 - 20

### DOI: 10.31029/demr.8.2

We consider the problem of inversion of the Laplace transform by means of a special series with respect to Laguerre polynomials, which in a particular case coincides with the Fourier series in polynomials $l_{r,k}^{\gamma}(x)$ $(r\in \mathbb{N}, k=0,1,\ldots)$, orthogonal with respect to a scalar product of Sobolev type of the following type
$<f,g>=\sum\nolimits_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_0^\infty f^{(r)}(t)g^{(r)}(t)t^\gamma e^{-t}dt, \gamma>-1.$
Estimates of the approximation of functions by partial sums of a special series with respect to Laguerre polynomials are given.

**Keywords: **
Laplace transforms, Laguerre polynomials, special series.