Abstract
We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting map**s in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.
Similar content being viewed by others
References
N. K. Karapetyants and S. G. Samko, Equations with Involutive Operators and their Applications (Rostov Gos. Univ., Rostov-on-Don, 1988) [in Russian].
N. I. Akhiezer, Elements of the Theory of Elliptic Functions, 2ns ed. (Nauka, Moscow, 1979) [in Russian].
R. V. Duduchava, “Convolution integral equations with discontinuous presymbols, singular integral equations with fixed singularities, and their applications to problems of mechanics,” Tr. Tbil. Mat. Inst. AN GruzSSR 60, 1–134 (1974).
T. Carleman, “Sur la théorie des equations intégrals et ses applications,” in Proceedings of the International Mathematical Congress, Zurich (1932), Vol. 1, pp. 138–151.
F. N. Garif’yanov, “On biorthogonal systems generated by some involutive operators,” Russ. Math. (Iz. VUZ) 47 (10), 24–34 (2003).
F. N. Garif’yanov and S. A. Modina, “The Carleman kernel and its applications,” Siber. Math. J. 53, 1011–1020 (2012).
E. P. Aksent’eva and F. N. Garif’yanov, “On the investigation of an integral equation with a Carleman kernel,” Sov. Math.(Iz.VUZ) 27 (4), 53–63 (1983).
E. I. Zverovich, “Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces,” Russ. Math. Surv. 26, 117–192 (1971).
A. F. Leont’ev, Series of Exponential Functions (Nauka, Moscow, 1976) [in Russian].
Yu.F. Korobeinik, “Representing systems,” Russ. Math. Surv. 36, 75–137 (1981).
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Garif’yanov, F.N., Strezhneva, E.V. Biorthogonal Systems of Analytic Functions Generated by a Regular Triangle. Lobachevskii J Math 40, 1275–1282 (2019). https://doi.org/10.1134/S1995080219090075
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080219090075