Cauchy type problems and boundary value problems in the complex domain (the case of even segments)

Abstract

AbstractThe results of this chapter are similar to those obtained in Chapter 11, but the problems considered here are essentially different. Namely, the Cauchy type problems considered here are formulated in terms of another pair of associated integrodifferential operators —$$ {\mathbb{L}_s} and \mathbb{L}_s^*$$ (where s ≥1 is any integer), and the corresponding boundary conditions are assumed to be satisfied at the endpoints of the sum of even (2s) segments\(\gamma {2_s}(\sigma ) = \bigcup\limits_{h = 0}^{2s - 1} \{ z = r\exp [i\pi (h + 1/2)]:0 \leqslant r \leqslant \sigma \}\) in the Riemann surface G∞ of Lnz. Using the results of Chapters 8 and 9 we prove the main Theorem 12.4-1 on the basis property of certain systems of functions in.$${L_2}\{ {\gamma _{2s}}(\sigma ))\}.$$ The consequence of this theorem is the important Theorem 12.4-2 on expansions of functions of $${L_2}\{ {\gamma _{2s}}(\sigma )\}$$ in terms of the systems of eigenfunctions and adjoint functions of the first of two boundary value problems considered here. Another consequence of Theorem 12.4-1 is Theorem 12.4-3 containing the construction of some systems of entire functions which are bases of weighted spaces L2 over the sum of segments\( {\Gamma _{2s}}(\sigma ) = \bigcup\limits_{h = 0}^{2s - 1} {\left\{ {z = r\exp [i\pi (h + 1/2)/s]:0 \leqslant r \leqslant {\sigma ^{1/s}}} \right\}} \subset \mathbb{C}.\)KeywordsEntire FunctionVector FunctionComplex DomainRiesz BaseCommon EndpointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.