Title: Computable Frames in Computable Banach Spaces
Author(s): S.K. Kaushik, Poonam Mantry
Pages: 93-100
Cite as:
S.K. Kaushik, Poonam Mantry, Computable Frames in Computable Banach Spaces, Int. J. Anal. Appl., 11 (2) (2016), 93-100.

Abstract


We develop some parts of the frame theory in Banach spaces from the point of view of Computable Analysis. We define computable M-basis and use it to construct a computable Banach space of scalar valued sequences. Computable Xd frames and computable Banach frames are also defined and computable versions of sufficient conditions for their existence are obtained.

Full Text: PDF

 

References


  1. S. Banach and S. Mazur, Sur les fonctions calculables, Ann. Soc. Pol. de Math. 16 (1937), 223.

  2. V. Brattka, Computability Of Banach Space Principles, Informatik Berichte 286, FernUniversitat Hagen, Fachbereich Informatik, Hagen, June 2001.

  3. V. Brattka, Effective Representations of the space of linear bounded operators, Applied General Topology, 4(1)(2003), 115-131.

  4. V. Brattka, Computable versions of the uniform boundedness Theorem, in: Z. Chatzidakis, P.Koepke, W. Pohlers (Eds.), Logic Colloquium 2002, in: Lecture Notes in Logic, Vol 27, Association for Symbolic Logic, Urbana, 2006.

  5. V. Brattka, Computability over topological structures. In S. Barry Cooper and Sergey S. Goncharov, editors, Computability and Models, 93-136. Kluwer Academic Publishers, New York, 2003.

  6. V. Brattka, A computable version of Banach’s Inverse Mapping Theorem, Annals of Pure and Applied Logic, 157 (2009), 85-96.

  7. V. Brattka, R. Dillhage, Computability of compact operators on computable Banach spaces with bases. Math. Log. Quart.53 (2007), 345-364.

  8. V. Brattka, A. Yoshikawa, Towards computability of elliptic boundary value problems in variational formulation, J. Complexity 22(6) (2006), 858-880.

  9. P.G. Casazza, O. Christensen, D.T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl., 307 (2005), 710-723.

  10. K. Grochenig, Describing functions : Atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1-41.

  11. A. Grzegorczyk, On the definitions of computable real continuous functions, Fund. Math. 44 (1957), 61-71.

  12. C. Kreitz, K. Weihrauch, A unified approach to constructive and recursive analysis, in: M. Richter, E. Borger, W. Oberschelp, B.Schinzel, W. Thomas (Eds.), Computation and Proof Theory, Lecture Notes in Mathematics, Vol. 1104, Springer, Berlin, 1984, 259-278.

  13. D. Lacombe, Les ensembles recursivement ouverts ou fermes, et leurs applications a 1’Analyse recursive, Comptes, Rendus 246 (1958), 28-31.

  14. D. Lacombe, Quelques procedes de definition en topologie recursive, in: A. Heyting, editor, Constructivity in mathematics, (North- Holland, Amsterdam 1959), 129-158.

  15. R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics 183, Springer, New York, 1989.

  16. M.B. Pour-El, J.I. Richards, Computability in Analysis and Physics, Springer, Berlin, 1989.

  17. I. Singer, Bases in Banach spaces-II, Springer-Verlag, New York, Heidelberg, 1981.

  18. A. M. Turing, On computable numbers, with an application to the ”Entscheidungsproblem”, Proc. London Math. Soc. 42 (1936), 230-265.

  19. K. Weihrauch, Computable Analysis, Springer, Berlin, 2000.