On the Frame Properties of System of Exponents with Piecewise Continuous Phase

Affiliation(s)

Institute of Mathematics and Mechanics of NASA, Baku, Azerbaijan.

Nakhchivan State University, Nakhchivan, Azerbaijan.

Institute of Mathematics and Mechanics of NASA, Baku, Azerbaijan.

Nakhchivan State University, Nakhchivan, Azerbaijan.

ABSTRACT

A double system of exponents with piecewise continuous complex-valued coefficients are considered. Under definite conditions on the coefficients the frame property of this system in Lebesgue spaces of functions is investigated. Such systems arise in the spectral problems for discontinuous differential operators.

Cite this paper

S. Farahani and T. Najafov, "On the Frame Properties of System of Exponents with Piecewise Continuous Phase,"*Applied Mathematics*, Vol. 4 No. 5, 2013, pp. 848-853. doi: 10.4236/am.2013.45116.

S. Farahani and T. Najafov, "On the Frame Properties of System of Exponents with Piecewise Continuous Phase,"

References

[1] R. Paley and N. Wiener, “Fourier Transforms in the Complex Domain,” American Mathematical Society, Providence, 1934.

[2] N. Levinson, “Gap and Density Theorems,” American Mathematical Society, Providence, 1940.

[3] R. M. Young, “An Introduction to Non-Harmonic Fourier Series,” Springer, Berlin, 1980, p. 246.

[4] A. M. Sedletskii, “Classes of Analytic Fourier Transformations and Exponential Approximations,” Fizmatlit, Moscow, 2005.

[5] Ch. Heil, “A Basis Theory Primer,” Springer, Berlin, 2011, p. 534. doi:10.1007/978-0-8176-4687-5

[6] O. Christensen, “An Introduction to Frames and Riesz bases,” Springer, Berlin, 2003, p. 440.

[7] D. L. Russell, “On Exponential Bases for the Sobolev Spaces Over an Interval,” Journal of Mathematical Analysis and Applications, Vol. 87, No. 2, 1982, pp. 528-550. doi:10.1016/0022-247X(82)90142-1

[8] X. He and H. Volkmer, “Riesz Bases of Solutions of Sturm-Lioville Equations,” Journal of Fourier Analysis and Applications, Vol. 7, No. 3, 2001, pp. 297-307. doi:10.1007/BF02511815

[9] H. Miklos, “Inverse Spectral Problems and Closed Ex ponential Systems,” Annals of Mathematics, Vol. 162, No. 2, 2005, pp. 885-918. doi:10.4007/annals.2005.162.885

[10] A. M. Sedletskii, “Nonharmonic Analysis,” Functional Analysis, Itogi Nauki i Tekhniki Seremennaya Matematika iee Prilozheniya Tematicheskie Obzory, Vol. 96, 2006, pp. 106-211.

[11] L. H. Larsen, “Internal Waves Incident upon a Knife Edge Barrier,” Deep Sea Research, Vol. 16, No. 5, 1969, pp. 411-419.

[12] S. A. Gabov and P. A. Krutitskii, “On Larsen’s Non stationary Problem,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, Vol. 27, No. 8, 1987, pp. 1184-1194.

[13] P. A. Krutitskii, “Small Non-Stationary Vibrations of Vertical Plates in a Channel with a Stratified Fluid,” USSR Computational Mathematics and Mathematical Physics, Vol. 28, No. 6, 1988, pp. 166-176.

[14] E. I. Moiseev and N. Abbasi, “Basis Property of Eigen functions of the Generalized Gasedynamic Problem of Frankl with a Nonlocal Oddness Condition and with the Discontinuity of the Gradient of Solution,” Differential Equations, Vol. 45, No. 10, 2009, pp. 1452-1456.

[15] V. A. Ilin, “Mixed Problem Describing the Damping Process of a Bar Consisting of Two Sections of Different Density and Elasticity Provided that the Time of Wave’s Passage in Each of These Sections Coincide,” Trudi Ins tituta Matematiki i Mekhaniki Uro RAN, Vol. 269, 2010, pp. 132-141.

[16] I. S. Lomov, “Non-Smooth Eigenfunctions in Problems of Mathematical Physics,” Differential Equations, Vol. 47, No. 3, 2011, pp. 358-365.

[17] L. M. Lujina, “Regularity of Spectral Problems with Additional Conditions at the Inner Points,” Matematiches kie Zametki, Vol. 49, No. 3, 1991, pp. 151-153.

[18] B. T. Bilalov and S. M. Farahani, “On Perturbed Bases of Exponential Functions with Complex Coefficients,” Trans actions of NAS of Azerbaijan, Vol. 56, No. 4, 2011, pp. 45-50.

[19] I. Singer, “Bases in Banach Spaces, I,” Springer, Berlin, 1970, p. 673. doi:10.1007/978-3-642-51633-7

[20] I. T. Hochberg and A. S. Markus, “On Stability of Bases of Banach and Hilbert Spaces,” Izvestiya Akademii Nauk Moldavskoj SSR, No. 5, 1962, pp. 17-35.

[21] B. T. Bilalov and T. R. Muradov, “On Equivalent Bases in Banach Spaces,” Ukrainian Mathematical Journal, Vol. 59, No. 4, 2007, pp. 551-554. doi:10.1007/s11253-007-0040-1

[22] B. T. Bilalov, “Bases from Exponents, Cosines and Sines Being Eigen Functions of Differential Operators,” Differential Equations, Vol. 39, No. 5, 2003, pp. 1-5.

[23] B. T. Bilalov, “Basicity of Some Systems of Exponents, Cosines and Sines,” Differential Equations, Vol. 20, No. 1, 1990, pp. 10-16.

[24] B. T. Bilalov, “On Isomorphism of Two Bases,” Funda mentalnaya i Prikladnaya Matematika, Vol. 1, No. 4, 1995, pp. 1091-1094.

[1] R. Paley and N. Wiener, “Fourier Transforms in the Complex Domain,” American Mathematical Society, Providence, 1934.

[2] N. Levinson, “Gap and Density Theorems,” American Mathematical Society, Providence, 1940.

[3] R. M. Young, “An Introduction to Non-Harmonic Fourier Series,” Springer, Berlin, 1980, p. 246.

[4] A. M. Sedletskii, “Classes of Analytic Fourier Transformations and Exponential Approximations,” Fizmatlit, Moscow, 2005.

[5] Ch. Heil, “A Basis Theory Primer,” Springer, Berlin, 2011, p. 534. doi:10.1007/978-0-8176-4687-5

[6] O. Christensen, “An Introduction to Frames and Riesz bases,” Springer, Berlin, 2003, p. 440.

[7] D. L. Russell, “On Exponential Bases for the Sobolev Spaces Over an Interval,” Journal of Mathematical Analysis and Applications, Vol. 87, No. 2, 1982, pp. 528-550. doi:10.1016/0022-247X(82)90142-1

[8] X. He and H. Volkmer, “Riesz Bases of Solutions of Sturm-Lioville Equations,” Journal of Fourier Analysis and Applications, Vol. 7, No. 3, 2001, pp. 297-307. doi:10.1007/BF02511815

[9] H. Miklos, “Inverse Spectral Problems and Closed Ex ponential Systems,” Annals of Mathematics, Vol. 162, No. 2, 2005, pp. 885-918. doi:10.4007/annals.2005.162.885

[10] A. M. Sedletskii, “Nonharmonic Analysis,” Functional Analysis, Itogi Nauki i Tekhniki Seremennaya Matematika iee Prilozheniya Tematicheskie Obzory, Vol. 96, 2006, pp. 106-211.

[11] L. H. Larsen, “Internal Waves Incident upon a Knife Edge Barrier,” Deep Sea Research, Vol. 16, No. 5, 1969, pp. 411-419.

[12] S. A. Gabov and P. A. Krutitskii, “On Larsen’s Non stationary Problem,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, Vol. 27, No. 8, 1987, pp. 1184-1194.

[13] P. A. Krutitskii, “Small Non-Stationary Vibrations of Vertical Plates in a Channel with a Stratified Fluid,” USSR Computational Mathematics and Mathematical Physics, Vol. 28, No. 6, 1988, pp. 166-176.

[14] E. I. Moiseev and N. Abbasi, “Basis Property of Eigen functions of the Generalized Gasedynamic Problem of Frankl with a Nonlocal Oddness Condition and with the Discontinuity of the Gradient of Solution,” Differential Equations, Vol. 45, No. 10, 2009, pp. 1452-1456.

[15] V. A. Ilin, “Mixed Problem Describing the Damping Process of a Bar Consisting of Two Sections of Different Density and Elasticity Provided that the Time of Wave’s Passage in Each of These Sections Coincide,” Trudi Ins tituta Matematiki i Mekhaniki Uro RAN, Vol. 269, 2010, pp. 132-141.

[16] I. S. Lomov, “Non-Smooth Eigenfunctions in Problems of Mathematical Physics,” Differential Equations, Vol. 47, No. 3, 2011, pp. 358-365.

[17] L. M. Lujina, “Regularity of Spectral Problems with Additional Conditions at the Inner Points,” Matematiches kie Zametki, Vol. 49, No. 3, 1991, pp. 151-153.

[18] B. T. Bilalov and S. M. Farahani, “On Perturbed Bases of Exponential Functions with Complex Coefficients,” Trans actions of NAS of Azerbaijan, Vol. 56, No. 4, 2011, pp. 45-50.

[19] I. Singer, “Bases in Banach Spaces, I,” Springer, Berlin, 1970, p. 673. doi:10.1007/978-3-642-51633-7

[20] I. T. Hochberg and A. S. Markus, “On Stability of Bases of Banach and Hilbert Spaces,” Izvestiya Akademii Nauk Moldavskoj SSR, No. 5, 1962, pp. 17-35.

[21] B. T. Bilalov and T. R. Muradov, “On Equivalent Bases in Banach Spaces,” Ukrainian Mathematical Journal, Vol. 59, No. 4, 2007, pp. 551-554. doi:10.1007/s11253-007-0040-1

[22] B. T. Bilalov, “Bases from Exponents, Cosines and Sines Being Eigen Functions of Differential Operators,” Differential Equations, Vol. 39, No. 5, 2003, pp. 1-5.

[23] B. T. Bilalov, “Basicity of Some Systems of Exponents, Cosines and Sines,” Differential Equations, Vol. 20, No. 1, 1990, pp. 10-16.

[24] B. T. Bilalov, “On Isomorphism of Two Bases,” Funda mentalnaya i Prikladnaya Matematika, Vol. 1, No. 4, 1995, pp. 1091-1094.