Communications in Mathematical Analysis

On the completeness of systems of recursive integrals

Vladislav V. Kravchenko

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Given a sufficiently smooth, nonvanishing complex valued function of a real variable defined on a finite interval, we show that it generates a complete system of functions in the $L_{2}$-space constructed by means of a recursive integration procedure.

Article information

Commun. Math. Anal., Conference 3 (2011), 172-176.

First available in Project Euclid: 25 February 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B24: Sturm-Liouville theory [See also 34Lxx]
Secondary: 41A30: Approximation by other special function classes 42A65: Completeness of sets of functions

Recursive integrals Sturm-Liouville problem complete system of functions.


Kravchenko, Vladislav V. On the completeness of systems of recursive integrals. Commun. Math. Anal. (2011), no. 3, 172--176.

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  • L. Bers, Theory of Pseudo-Analytic Functions. New York University 1952.
  • H. Campos and V. V. Kravchenko, A finite-sum representation for solutions for the Jacobi operator. Journal of Difference Equations and Applications. (2010). First published on: 05 March 2010 (iFirst). To link to this Article: DOI: 10.1080/10236190903158990.
  • R. Castillo-Pérez, K. V. Khmelnytskaya, V. V. Kravchenko, and H. Oviedo, Efficient calculation of the reflectance and transmittance of finite inhomogeneous layers. Journal of Optics A: Pure and Applied Optics 11 (2009), No. 6, pp 065707.
  • L. Collatz, Functional Analysis and Computational Mathematics. Mir, Moscow 1969 (Russian translation of the German edition Collatz L. Funktionalanalysis und Numerische Mathematik. Springer-Verlag, Berlin 1964).
  • Ph. J. Davis, Interpolation and Approximation. Dover Publications 1975.
  • K. V. Khmelnytskaya and H. C. Rosu, An amplitude-phase (Ermakov–Lewis) approach for the Jackiw–Pi model of bilayer graphene. J. of Phys. A 42 (2009), No. 4, pp 042004.
  • K. V. Khmelnytskaya and H. C. Rosu, A new series representation for Hill's discriminant. Annals of Physics (2010) doi: 10.1016/j.aop.2010.06.009.
  • K. V. Khmelnytskaya and H. C. Rosu, Bloch Solutions of Periodic Dirac Equations in SPPS Form. (2010) arXiv:1006.3596.
  • K. V. Khmelnytskaya, H. C. Rosu, and A. Gonzalez, Periodic Sturm-Liouville problems related to two Riccati equations of constant coefficients. Annals of Physics 33 (2010), issue 4, pp 469-472.
  • K. V. Khmelnytskaya and T. V. Torchynska, Reconstruction of potentials in quantum dots and other small symmetric structures. Mathematical Methods in the Applied Sciences 33 (2010), issue 4, pp 469-472.
  • V. V. Kravchenko, A representation for solutions of the Sturm-Liouville equation. Complex Variables and Elliptic Problems 53 (2008), issue 8, pp 775-789.
  • V. V. Kravchenko, Applied Pseudoanalytic Function Theory. Series: Frontiers in Mathematics. Birkhäuser, Basel 2009.
  • V. V. Kravchenko and R. M. Porter, Spectral parameter power series for Sturm-Liouville problems. Mathematical Methods in the Applied Sciences 33 (2010), issue 4, pp 459-468.
  • V. V. Kravchenko and R. M. Porter, Conformal Mapping of Right Circular Quadrilaterals. Complex Variables and Elliptic Equations (2010), First Published on: 08 July 2010. DOI: 10.1080/17476930903276100.
  • B. M. Levitan and I. S Sargsjan, Sturm-Liouville and Dirac operators. Kluwer Acad. Publ., Dordrecht 1991.
  • V. A. Marchenko, Sturm-Liouville Operators and ASpplications. Birkhäuser, Basel 1986.