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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.)
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