Sobolev orthogonal systems with two discrete points and Fourier series (Q2066438)

Denote by \(W_{L^2}^{1}=W_{L^2}^{1}[a, b]\) a Sobolev space consisting of absolutely continuous on \([a,b]\) functions \(f\) such that \(f^\prime\in L^2[a, b]\). Let space \(W_{L^2}^1\) be endowed with inner product \[ \langle f, g\rangle_{S} = f(a)g(a) + f(b)g(b) +\int_{a}^{b} f^\prime(t) g^\prime(t) dt. \] Let \(\Phi=\{\varphi_k\}_{k=0}^{\infty}\) be a system of functions from \(L^2([a,b])\) such that \[ \int_{a}^{b}\varphi_0(t)dt\not =0, \qquad \int_{a}^{b}\varphi_k(t)=0, \quad k\geq 1. \] Introduce a new system of functions \(\Phi_1=\{\varphi_{1,k}\}\) defined by: \begin{align*} \varphi_{1,0}(x)&=\frac{1}{\sqrt{2}}, \\ \varphi_{1,1}(x)&=\frac{1}{\sqrt{1+\frac{1}{2}J_0^2}}\left(-\frac{1}{2}J_0+\int_{a}^{x}\varphi_0(t)dt\right),\quad J_0=\int_{a}^{b}\varphi_0(t)dt, \\ \varphi_{1,k}(x)&=\int_{a}^{x}\varphi_{k-1}(t)dt,\quad k\geq 2. \end{align*} In the presented paper it is proved that if the system \(\Phi\) is a complete ortonormal system in \(L^2\) then \(\Phi_1\) is a complete orthonormal system in \(W_{L^2}^{1}\). It is also established that if the system \(\Phi_1\) is a complete orthonormal system in \(W_{L^2}^{1}\) then for any function \(f\in W_{L^2}^{1}\) its Fourier series with respect to the system \(\Phi_1\) uniformly converges to the function itself.