Fourier-Bessel-type series: the fourth-order case
The structured higher-order Bessel-type functions were introduced in 1994. They satisfy linear ordinary differential equations with analytic coefficients in the complex plane \(\mathbb{C}\), the so-called higher-order Bessel-type differential equations. There is a countable infinity of these equations, all of them of even order. These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases also bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth- and higher-order cases. This paper follows a number of earlier papers devoted to the study of the fourth-order case. On the positive half \([ 0,+ \infty )\) of the real line this differential equation has the Lagrange symmetric form \(L_M [ y ] (x) := \bigl( x y''(x) \bigr)'' - \bigl( ( 9x^{-1} + 8 M^{-1} x) y'(x) \bigr)' = \Lambda \;x \;y(x)\) for all \(x \in (0,\infty)\). Here \(M \in (0,\infty)\) is a given parameter and \(\Lambda \in \mathbb{C}\) is the spectral parameter. The spectral theory of this differential equation has been considered in certain weighted Lebesgue and Lebesgue-Stieltjes Hilbert function spaces. In this paper the spectral theory of the fourth-order Bessel-type differential equation is developed in the Lebesgue-Stieltjes space \(L^2([ 0,1 ] ; m_k )\) consisting of all Borel measurable functions \(f: [ 0,1 ] \rightarrow \mathbb{C}\) for which the norm and inner-product are given by \(\| f\|_k^2 = k \;|f(0)|^2 + \int_0^1 x \;|f(x)|^2 \;dx\) and \((f,g)_k = k f(0) \;\overline{g(0)} + \int_0^1 x \;f(x) \;\overline{g(x)} \;dx\) with \(k \in (0,\infty)\) a parameter. It is shown that for all values of the two parameters \(k,M \in (0,\infty)\) the self-adjoint operators generated by the Lagrange symmetric differential expression \(L_M\) in the space \(L^2([ 0,1 ] ; m_k )\) have a discrete simple spectrum and are bounded below. This study leads to the development of the properties of the Fourier-Bessel-type series for the fourth-order case. It turns out that for this particular purpose it is essential to make a special relationship between the parameters \(k\) and \(M\) by taking \(k=M/2\). This choice is essential for the determination of a self-adjoint operator generated by \(L_M\), the eigenvectors of which give eigensolutions solely dependent on the principal Bessel-type function \(J_{\lambda}^{0,M}\), thereby extending the classical theory of the second-order Fourier-Bessel series to the fourth-order case. In fact the Fourier-Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions \(J_{\lambda}^{0,M}\) of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.
scientific article; zbMATH DE number 5641721
scientific article