## On uniform convergence on closed intervals of spectral expansions and their derivatives, for functions from $$W_p^{(1)}$$.(English)Zbl 1095.34052

Let $$L$$ be a positive differential operator defined by the expression $$Lu(x):=-u''(x)+q(x)u(x)$$ on a bounded interval $$G\subset\mathbb R$$ and a pair of selfadjoint boundary conditions, where $$q\in L_1(G)$$ is a real function. Let $$\{u_n(x)\}_{n=1}^{\infty}$$ be a complete (in $$L_2(G)$$) and orthonormal system of eigenfunctions of $$L$$, and let $$\{\lambda_n\}_{n=1}^{\infty}$$ be the corresponding system of positive eigenvalues, enumerated in non-decreasing order. For $$f\in L_1(G)$$ and $$\mu>2$$, form the partial sum of order $$\mu$$ of the spectral expansion for $$f$$, $$\sigma_{\mu}(x,f)=\sum_{n=1}^{[\mu]}f_nu_n(x)$$, where $$f_n=\int_a^bf(x)u_n(x)\,dx$$. The author proves the following statements about the uniform convergence of this expansion and its derivatives.
1. If $$f\in W_p^{(1)}(G)$$, $$1<p\leq2$$ and $$f(a)=f(b)=0$$, then for $$x\in\overline G$$ the equality $$f(x)=\sum_{n=1}^{\infty}f_nu_n(x)$$ is valid, the series is absolutely and uniformly convergent on $$\overline G$$ and the estimate
$\max_{x\in\overline G}|f(x)-\sigma_{\mu}(x,f)|=o(1/\mu^{1-1/p}),\quad \mu\to\infty,$
holds.
2. If $$f\in D(L)$$, then for every $$x\in\overline G$$ and $$j=0,1$$, the equalities $$f^{(j)}(x)=\sum_{n=1}^{\infty}f_nu_n^{(j)}(x)$$ are valid, the series are absolutely and uniformly convergent on $$\overline G$$ and the estimates
$\max_{x\in\overline G}|f^{(j)}(x)-\sigma_{\mu}^{(j)}{\mu}(x,f)|=o(\mu^{3/2-j}),\quad \mu\to\infty,$
hold.
3. If $$q\in W_1^{(1)}(G)$$, $$f\in D(L)\cap W_1^{(3)}(G)$$, $$Lf\in W_p^{(1)}(G)$$, $$1<p\leq2$$ and $$Lf(a)=Lf(b)=0$$, then for every $$x\in\overline G$$ and $$j=0,1,2$$, the equalities $$f^{(j)}(x)=\sum_{n=1}^{\infty}f_nu_n^{(j)}(x)$$ are valid, the series are absolutely and uniformly convergent on $$\overline G$$ and the estimates
$\max_{x\in\overline G}|f^{(j)}(x)-\sigma_{\mu}^{(j)}{\mu}(x,f)|=o(\mu^{3-j-1/p}), \quad \mu\to\infty,$
hold.

### MSC:

 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 47E05 General theory of ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

### Keywords:

Schrödinger operator; classes $$W_p^{(1)}$$