Sampling theorem associated with a Dirac operator and the Hartley transform
A Kramer's type sampling theorem associated with a BVP of the form \[ y_1'+ r(x)y_2= -\lambda y_2,\quad y_2'- p(x)y_1= \lambda y_1,\tag{S} \] \[ y_1(0)\sin\alpha+ y_2(0)\cos\alpha= 0, \quad y_1(\pi)\sin\beta+ y_2(\pi)\cos\beta= 0\tag{BV} \] is proved. More exactly, the authors show the following: Let \(G(\lambda)\) be an entire complex valued function with zeros, the eigenvalues \(\lambda_n\), \(n\in\mathbb{Z}\) of (S)-(BV) and let \(\phi(x,\lambda)\) be a solution of (S) satisfying \(\phi(0,\lambda)^T= (\cos\alpha, -\sin\alpha)\), (upper index \(T\) denotes transposition). Then, given any \(f\in L^2([0,\pi], \mathbb{R}^2)\) the function \[ F(\lambda)= \int^\pi_0 f^T(x)\phi(x,\lambda)dx \] is an entire complex function of exponential type at most \(\pi\). Moreover, the series \[ \sum^{+\infty}_{n= -\infty} F(\lambda_n)= {G(\lambda)\over (\lambda-\lambda_n)G(\lambda_n)} \] converges almost uniformly on \(\mathbb{C}\) to the function \(F\).
scientific article; zbMATH DE number 1123608