Cite this article as:
Al-Jourany K. H., Mironov V. A., Terekhin P. A. Affine System of Walsh Type. Completeness and Minimality. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 3, pp. 247-256. DOI: https://doi.org/10.18500/1816-9791-2016-16-3-247-256
Affine System of Walsh Type. Completeness and Minimality
The question on completeness and minimality of Walsh affine systems is considered. On the basis of functional-analytical structure of multishift in Hilbert space, which being the generalized analogue of the operator of simple one-side shift and closely connected with Cuntz algebra representations, we give definition of Walsh affine system. Various criteria and tests of completeness of affine systems of functions are established. A biorthogonal conjugate system is found and its completeness is proved.
1. Terekhin P. A. On representation properties of a system of contractions and shifts of functions on an interval. Izv. Tul’sk. Gos. Univ., Ser. Matem., Mekh., Inform., 1998, vol. 4, no. 1, pp. 136–138 (in Russian).
2. Terekhin P. A. On the multiplicative structure of the centralizer of a multishift on a Hilbert space. Mathematics. Mechanics: Collection of Scientific Papers, Saratov, Saratov Univ. Press, 2000, iss. 2, pp. 119–122 (in Russian).
3. Terekhin P. A. Multishifts in Hilbert spaces. Funct. Anal. Appl., 2005, vol. 39, no. 1, pp. 57–67. DOI: https://doi.org/10.1007/s10688-005-0017-5.
4. Terekhin P. A. Affine Systems of Walsh Type. Orthogonalization and Completion. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 4, pt. 1, pp. 395–400 (in Russian).
5. Mironov V. A., Terekhin P. A. Minimality of an affine systems of Walsh type. Mathematics. Mechanics: Collection of Scientific Papers, Saratov, Saratov Univ. Press, 2014, iss. 16, pp. 41–44 (in Russian).
6. Mironov V. A., Terekhin P. A. Trigonometric affine system of Walsh type. Mathematics. Mechanics: Collection of Scientific Papers, Saratov, Saratov Univ. Press, 2015, iss. 17, pp. 37–39 (in Russian).
7. Filippov V. I., Oswald P. Reprentation in Lp by series of translates and dilates of one function. J. Approx. Theory, 1995, vol. 82, no. 1, pp. 15–29. DOI: https://doi.org/10.1006/jath.1995.1065.
8. Galatenko V. V., Lukashenko T. P., Sadovnichii V. A. On the properties of orthorecursive expansions with respect to subspaces. Proc. Steklov Inst. Math., 2014, vol. 284, pp. 129–132. DOI: https://doi.org/10.1134/S0081543814010076.
9. Kudryavtsev A. Yu. On the convergence of оrthorecursive expansions in nonorthogonal wavelets. Math. Notes, 2012, vol. 92, iss. 5, pp. 643–656. DOI: https://doi.org/10.1134/S0001434612110077.
10. Sil’nichenko A. V. On the convergence of orderpreserving weak greedy algorithms. Math. Notes, 2008, vol. 84, iss. 5, pp. 741–747. DOI: https://doi.org/10.1134/S0001434608110187.
11. Sarsenbi A. M., Terekhin P. A. Riesz basicity for general systems of functions. J. Function Spaces, 2014, vol. 2014, article ID 860279, pp. 1–3. DOI: https://doi.org/10.1155/2014/860279.
12. Terekhin P. A. Translates and dilates of function with nonzero integral. Mathematics. Mechanics: Collection of Scientific Papers, Saratov, Saratov Univ. Press, 1999, iss. 1, pp. 67–68 (in Russian).
13. Terekhin P. A. Inequalities for the components of summable functions and their representations by elements of a system of contractions and shifts. Russian Math., 1999, vol. 43, no. 8, pp. 70–77.
14. Terekhin P. A. Riesz bases generated by contractions and translations of a function on an interval. Math. Notes, 2002, vol. 72, iss. 4, pp. 505–518. DOI: https://doi.org/10.1023/A:1020536412809.
15. Terekhin P. A. On perturbations of the Haar system. Math. Notes, 2004, vol. 75, iss. 3, pp. 466– 469. DOI: https://doi.org/10.1023/B:MATN.0000023325.89390.66.
16. Terekhin P. A. Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces L p [0, 1]. Math. Notes, 2008, vol. 83, iss. 5, pp. 722–740. DOI: https://doi.org/10.1134/S000143460805009X.
17. Terekhin P. A. On the components of summable functions represented by elements of families of wavelet functions. Russian Math., 2008, vol. 52, no. 2, pp. 51–57. DOI: https://doi.org/10.1007/s11982-008-2008-3.
18. Terekhin P. A. Linear algorithms of affine synthesis in the Lebesgue space L 1 [0, 1]. Izv. Math., 2010, vol. 74, iss. 5, pp. 993–1022. DOI: https://doi.org/10.1070/IM2010v074n05ABEH002513.
19. Terekhin P. A. Best approximation of functions in L p by polynomials on affine system. Sb. Math., 2011, vol. 202, no. 2, pp. 279—306. DOI: https://doi.org/10.1070/SM2011v202n02ABEH004146.
