Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type
Bilender P. Allahverdiev; Hüseyin Tuna
Communications in Mathematics (2020)
- Volume: 28, Issue: 1, page 13-25
- ISSN: 1804-1388
Access Full Article
topAbstract
topHow to cite
topAllahverdiev, Bilender P., and Tuna, Hüseyin. "Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type." Communications in Mathematics 28.1 (2020): 13-25. <http://eudml.org/doc/297163>.
@article{Allahverdiev2020,
abstract = {In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.},
author = {Allahverdiev, Bilender P., Tuna, Hüseyin},
journal = {Communications in Mathematics},
keywords = {Hahn's Sturm-Liouville equation; spectral function; Parseval equality; spectral expansion},
language = {eng},
number = {1},
pages = {13-25},
publisher = {University of Ostrava},
title = {Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type},
url = {http://eudml.org/doc/297163},
volume = {28},
year = {2020},
}
TY - JOUR
AU - Allahverdiev, Bilender P.
AU - Tuna, Hüseyin
TI - Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type
JO - Communications in Mathematics
PY - 2020
PB - University of Ostrava
VL - 28
IS - 1
SP - 13
EP - 25
AB - In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.
LA - eng
KW - Hahn's Sturm-Liouville equation; spectral function; Parseval equality; spectral expansion
UR - http://eudml.org/doc/297163
ER -
References
top- Aldwoah, K.A., Generalized time scales and associated difference equations, 2009, Cairo University, Ph.D. Thesis. (2009)
- Allahverdiev, B.P., Tuna, H., An expansion theorem for -Sturm-Liouville operators on the whole line, Turkish J. Math., 42, 3, 2018, 1060-1071, (2018) MR3804971
- Allahverdiev, B.P., Tuna, H., 10.3906/mat-1803-79, Turkish J. Math., 42, 5, 2018, 2527-2545, (2018) MR3866169DOI10.3906/mat-1803-79
- Allahverdiev, B.P., Tuna, H., Eigenfunction expansion in the singular case for Dirac systems on time scales, Konuralp J. Math., 7, 1, 2019, 128-135, (2019) MR3948622
- Allahverdiev, B.P., Tuna, H., 10.3906/mat-1902-16, Turkish J. Math., 43, 2019, 1668-1687, (2019) MR3962557DOI10.3906/mat-1902-16
- Allahverdiev, B.P., Tuna, H., Eigenfunction expansion for singular Sturm-Liouville problems with transmission conditions, Electron. J. Differ.Equat., 2019, 3, 2019, 1-10, (2019) MR3904844
- Allahverdiev, B.P., Tuna, H., The Parseval equality and expansion formula for singular Hahn-Dirac system, Emerging Applications of Differential Equations and Game Theory, 2020, 209-235, IGI Global, (2020)
- Álvarez-Nodarse, R., 10.1016/j.cam.2005.06.046, J. Comput. Appl. Math., 196, 1, 2006, 320-337, (2006) MR2241592DOI10.1016/j.cam.2005.06.046
- Annaby, M.H., Hamza, A.E., Aldwoah, K.A., 10.1007/s10957-012-9987-7, J. Optim. Theory Appl., 154, 2012, 133-153, (2012) MR2931371DOI10.1007/s10957-012-9987-7
- Annaby, M.H., Hamza, A.E., Makharesh, S.D., A Sturm-Liouville theory for Hahn difference operator, Frontiers of Orthogonal Polynomials and -Series, 2018, 35-84, World Scientific, Singapore, (2018) MR3791609
- Annaby, M.A., Hassan, H.A., 10.1016/j.jmaa.2018.04.016, J. Math. Anal. Appl., 464, 1, 2018, 493-506, (2018) MR3794101DOI10.1016/j.jmaa.2018.04.016
- Arvesú, J., 10.1016/j.cam.2009.02.062, J. Comput. Appl. Math., 233, 6, 2010, 1462-1469, Elsevier, doi:10.1016/j.cam.2009.02.062. (2010) MR2559332DOI10.1016/j.cam.2009.02.062
- Berezanskii, J.M., Expansions in Eigenfunctions of Selfadjoint Operators, 1968, Amer. Math. Soc., Providence, (1968) Zbl0157.16601MR0222718
- Dobrogowska, A., Odzijewicz, A., 10.1016/j.cam.2005.06.009, J. Comput. Appl. Math., 193, 1, 2006, 319-346, (2006) MR2228721DOI10.1016/j.cam.2005.06.009
- Guseinov, G.Sh., Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int. J. Difference Equat., 2, 1, 2007, 93-104, (2007) MR2374102
- Guseinov, G.Sh., An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales, Adv. Dyn. Syst. Appl., 3, 1, 2008, 147-160, (2008) MR2547666
- Hahn, W., 10.1002/mana.19490020103, Math. Nachr., 2, 1949, 4-34, (1949) MR0030647DOI10.1002/mana.19490020103
- Hahn, W., 10.1007/BF01301144, Monatsh. Math., 95, 1983, 19-24, (1983) MR0697345DOI10.1007/BF01301144
- Hamza, A.E., Ahmed, S.A., Existence and uniqueness of solutions of Hahn difference equations, Adv. Difference Equat., 316, 2013, 1-15, (2013) MR3337265
- Hamza, A.E., Makharesh, S.D., 10.24297/jam.v12i6.3836, J. Adv. Math., 12, 6, 2016, 6335-6345, (2016) DOI10.24297/jam.v12i6.3836
- Huseynov, A., Bairamov, E., On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9, 1, 2009, 77-88, (2009) MR2510666
- Huseynov, A., 10.1016/S0034-4877(10)00026-1, Rep. Math. Phys., 66, 2, 2010, 207-235, (2010) MR2777355DOI10.1016/S0034-4877(10)00026-1
- Jackson, F.H., 10.2307/2370183, Amer. J. Math., 32, 1910, 305-314, (1910) MR1506108DOI10.2307/2370183
- Jagerman, D.L., Difference Equations with Applications to Queues, 2000, Dekker, New York, (2000) MR1792377
- Jordan, C., Calculus of Finite Differences, 3rd edn, 1965, Chelsea, New York, (1965) MR0183987
- Kolmogorov, A.N., Fomin, S.V., Introductory Real Analysis. Translated by R.A. Silverman, 1970, Dover Publications, New York, (1970) MR0377445
- Kwon, K.H., Lee, D.W., Park, S.B., Yoo, B.H., Hahn class orthogonal polynomials, Kyungpook Math. J., 38, 1998, 259-281, (1998) MR1665852
- Lesky, P.A., Eine Charakterisierung der klassischen kontinuierlichen, diskretenund -Orthgonalpolynome, 2005, Shaker, Aachen, (2005)
- Levinson, N., 10.1215/S0012-7094-51-01806-6, Duke Math. J., 18, 1951, 57-71, (1951) MR0041313DOI10.1215/S0012-7094-51-01806-6
- Levitan, B.M., Sargsjan, I.S., Sturm-Liouville and Dirac Operators, 1991, Springer, (1991) MR1136037
- Naimark, M.A., Linear Differential Operators, 2nd edn., 1968, 1969, Nauka, Moscow, English translation of 1st edn.. (1969) MR0353061
- Petronilho, J., 10.1016/j.cam.2006.05.005, J. Comput. Appl. Math., 205, 2007, 314-324, (2007) MR2324843DOI10.1016/j.cam.2006.05.005
- Sitthiwirattham, T., On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different -derivatives, Adv. Difference Equat., 2016, 1, 2016, Article number 116. (2016) MR3490997
- Stone, M.H., 10.1090/S0002-9947-1926-1501372-6, Trans. Amer. Math. Soc., 28, 1926, 695-761, (1926) MR1501372DOI10.1090/S0002-9947-1926-1501372-6
- Stone, M.H., Linear Transformations in Hilbert Space and Their Application to Analysis, 1932, Amer. Math. Soc., (1932) MR1451877
- Titchmarsh, E.C., Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition, 1962, Clarendon Press, Oxford, (1962) MR0176151
- Weyl, H., 10.1007/BF01474161, Math. Annal., 68, 1910, 220-269, (1910) MR1511560DOI10.1007/BF01474161
- Yosida, K., 10.1017/S0027763000022820, Nagoya Math. J., 1, 1950, 49-58, (1950) MR0042016DOI10.1017/S0027763000022820
- Yosida, K., Lectures on Differential and Integral Equations, 1960, Springer, New York, (1960) MR0118869
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.