Bases of exponentials in the spaces \(L^p(-\pi,\pi)\) (Q1810203)

The author investigates the system of exponentials in \(L^p=L^p(\pi,\pi)\) \[ \left\{e^{i\lambda_nt},ite^{i\lambda_nt},\dots, (it)^{m_n-1}e^{i\lambda_nt}\right\}_{n=0}^{\infty}, \tag{1} \] where \( \left\{\lambda_n\right\}_{n=0}^{\infty}\) is the sequence of zeros of the entire function \[ L(z)=\int_{-\pi}^\pi e^{izt}\frac{k(t)\,dt}{(\pi^2-t^2)^\alpha},\quad0<\text{Re}\,\alpha<1, \quad\text{var}\,k<\infty,\quad k(\pm\pi\mp0)\neq0, \] \(m_n\) is the multiplicity of the root \(\lambda_n\), \(| \lambda_{n+1}| \geq| \lambda_{n}| \). For the system (1), the author proves an analog of the Riesz theorem on the projection from \(L^p\) onto \(H^p\). We say that the basis (1) of the space \(L^p \) (or one of its subspaces \(B\)) possesses the Riesz property if the projection \(\sum_{n=0}^\infty P_{m_n-1}(t)e^{i\lambda_nt}\to\sum_{\text{Re}\, \lambda_n\geq0} P_{m_n-1}(t)e^{i\lambda_nt}\) is bounded in \(L^p(B)\) ( \(P_m(t)\) is a polynomial of degree \(\leq m\)). The following assertions are valid: 1) for \(1<p<1/(1- \text{Re}\,\alpha)\), the system (1) forms a basis of \(L^p\) possessing the Riesz property; 2) for \(1/(1- \text{Re}\,\alpha) < p < \infty\), the system (1) forms a basis of \(L_0^p\) possessing the Riesz property (here \(L_0\) is the subspace in \(L^p\) annihilated by the measure \(\frac{k(t)\,dt}{(\pi^2-t^2)^\alpha})\). If \(1/(1-\text{Re}\,\alpha) < p < \infty\), then the deficiency of system (1) in \(L^p\) is equal to one. Therefore, the assertion 2) can be restated as follows: if \(\mu\neq\lambda_n\), then the system (1), augmented by the function \(e^{i\mu t}\), forms a basis of \(L^p\), \(1/(1-\text{Re}\,\alpha)<p<\infty\), possessing the Riesz property.