Ulam Stability Problem for Frames

Part of the Springer Optimization and Its Applications book series (SOIA, volume 52)


In this paper we give a solution to the Ulam stability problem for continuous Parseval frames in finite dimensional Hilbert spaces. We prove that if F is a nearly Parseval frame then there exists a Parseval frame near F. Also, we give generalizations of this result.


Hyers–Ulam-Rassias stability Frames 



The first author was supported by the Grant: POSDRU/88/1.5/S/49516, project: “Creşterea calităţii şi competivităţii cercetării doctorale prin acordarea de burse.”

The authors thanks Professors J. Brzdȩk, P.G. Casazza, O. Christensen and Th.M. Rassias for carefully reading and useful comments.


  1. 1.
    Ali, S.T., Antoine, J.P., Gazeau, J. P.: Continuous frames in Hilbert spaces. Ann. Physics 222, 1–37 (1993)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bodmann, B., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258, 397–420 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brzdȩk, J.: On a method of proving the Hyers–Ulam stability of functional equations on restricted domains. Aust. J. Math. Anal. Appl. 6, Issue 1, Article 4, 1–10 (2009)Google Scholar
  4. 4.
    Casazza, P.G., Kutyniok, G.: A generalization of Gram–Schmidt orthogonalization generating all Parseval frames. Adv. Comput. Math. 27, 65–78 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Casazza, P.G.: Local theory of frames and Schauder bases for Hilbert spaces. Illinois J. Math. 43, 291–306 (1999)MathSciNetMATHGoogle Scholar
  6. 6.
    Casazza, P.G.: Custom building finite frames. Contemp. Math. 345, 61–86 (2004)MathSciNetGoogle Scholar
  7. 7.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhaüser, Boston (2003)MATHGoogle Scholar
  8. 8.
    Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, New Jersey (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fornasier, M., Rauhut, H.: Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl. 11, 245–287 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gabardo, J.P., Han, D.: Frames associated with measurable space. Adv. Comput. Math. 18, 127–147 (2003)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Găvruţa, L., Găvruţa, P., Zamani Eskandani, G.: Hyers–Ulam stability of frames in Hilbert spaces. Bul. Ştiinţ. Univ. Politeh. Timiş. Ser. Mat. Fiz. 55 (69) 2, 60–67 (2010)Google Scholar
  14. 14.
    Găvruţa, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Găvruţa, P.: On some identities and inequalities for frames in Hilbert spaces. J. Math. Anal. Appl. 321, 469–478 (2006)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Găvruţa, P.: On the duality of fusion frames. J. Math. Anal. Appl. 333, 871–879 (2007)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Găvruţa, P.: On the wavelet transform and chirps. In: Proceedings of the 7th Symposium of Mathematics and its Applications, pp. 117–122, “Politehnica” University of Timisoara (1997)Google Scholar
  18. 18.
    Găvruţa, P., Găvruţa, L.: A new method for the generalized Hyers-Ulam-Rassias stability. Int. J. Nonlinear Anal. Appl. 1, no. 2, 11–18 (2010)Google Scholar
  19. 19.
    Găvruţa, P., Ciurdariu, L., Găvruţa, L.: On the Hyers-Ulam stability of Parseval frames. Bul. Ştiinţ. Univ. Politeh. Timiş. Ser. Mat. Fiz. 51 (65) 2, 12–17 (2006)Google Scholar
  20. 20.
    Han, D., Kornelson, K., Larson, D., Weber, E.: Frames for undergraduates. Amer. Math. Soc., SML/40, Providence, Rhode Island (2007)Google Scholar
  21. 21.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Soc. USA 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser (1998)Google Scholar
  23. 23.
    Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Florida (2001)MATHGoogle Scholar
  24. 24.
    Kaiser, G.: Quantum Physics, Relativity and Complex Spacetime. North-Holland Mathematics Studies, Vol. 163, Amsterdam (1990)Google Scholar
  25. 25.
    Kaiser, G.: A Friendly Guide to Wavelets. Birkhäuser, Boston (1994)MATHGoogle Scholar
  26. 26.
    Rahimi, A., Najati, A., Dehgan, Y.N.: Continuous frame in Hilbert space. Methods Funct. Anal. Topology 12, 170–182 (2006)MathSciNetMATHGoogle Scholar
  27. 27.
    Rassias, J.M.: Solution of a problem of Ulam. J. Approx. Theory 57, 268–273 (1989)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Rassias, Th.M.: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62, 23–130 (2000)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Ulam, S.M.: A Collection of Mathematical Problems Interscience Publishers, Inc., New York (1968)Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics“Politehnica” University of TimişoaraTimişoaraRomania

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