On representation of a solution to the Cauchy problem by a Fourier series in Sobolev-orthogonal polynomials generated by Laguerre polynomials
In this contribution, sequences of polynomials \(\{l^{\alpha}_{r,k}(x)\}_{k\geq 0}\) orthonormal with respect to the Sobolev inner product \((f,g)= \sum_{k=0}^{r-1} f^{(k)}(0)g^{(k)}(0)+ \int_{0}^{\infty} f^{(r)}(x) g^{(r)}(x) x^{\alpha} e^{-x} dx, \alpha >-1, \) are studied. They are given in terms of an integral representation involving the sequence of orthonormal Laguerre polynomials \(\{l^{\alpha}_{n}(x)\}_{n\geq 0}\). Indeed, \(l^{\alpha}_{r,k}(x)= \frac{x^{k}}{k!}, 0\leq k\leq r-1,\) and \(l^{\alpha}_{r,r+k}(x)= \frac{1}{(r-1)!}\int_{0}^{x} (x-t)^{r-1} l^{\alpha}_{k}(t)dt, k\geq 0\). In Theorem 3.1 is proved that they constitute a complete orthonormal system in the space \(W^{r}_{ 2}\) of the class of continuously differentiable \(r-1\) times functions such that its \((r-1)\)-th order derivative is absolutely continuous on an arbitrary segment \([a,b] \subset [0, \infty)\) and its \(r\)-th order derivative belongs to the weighted \(L^{2}\) space defined by the Gamma weight. Moreover, for \(-1<\alpha<1 ,\) the Fourier series, in terms of the above orthonormal polynomials, of a function \(f\in W^{r}_{2}\) uniformly converges in \([0,A]\) as well as pointwise in \([0, \infty)\). Notice that the constraint about the values of \(\alpha\) is a sufficient condition for convergence. The rate of convergence of the Sobolev-Fourier projector is analyzed in the particular case \(\alpha=0\). Finally, a representation of the solution of a Cauchy problem for an ODE as a Fourier series with respect to the above orthonormal basis is given. Notice that when the Cauchy problem is solved by a spectral method by using the Fourier expansion of the solution, the problem of seeking is reduced to determine the Fourier coefficients of the above expansion. When the differential equation is linear, then the problem is reduced to solve a linear system of equations. In the case \(\alpha=0\), the Fourier coefficients are determined recurrently from a very simple system of equations.
Sobolev inner products and orthogonal polynomials
classical orthogonal Laguerre polynomials
scientific article; zbMATH DE number 6899062
scientific article