Stability of the bases and frames reproducing kernels in model spaces  [ Stabilité de bases et frames des noyaux reproduisants dans les espaces modèles ]
Annales de l'Institut Fourier, Tome 55 (2005) no. 7, p. 2399-2422
On étudie les bases et les frames des noyaux reproduisants dans les sous-espaces modèles K Θ 2 =H 2 ΘH 2 de l’espace de Hardy H 2 dans le demi-plan supérieur. On considère le problème de la stabilité d’une base des noyaux reproduisants k λ n (z)=(1-Θ(λ n ) ¯Θ(z))/(z-λ ¯ n ) par rapport aux petites perturbations des pôles λ ¯ n . En utilisant les majorations récentes des derivées dans les espaces K Θ 2 , on obtient les estimations des perturbations admissibles, qui généralisent les théorèmes de W.S. Cohn et E. Fricain.
We study the bases and frames of reproducing kernels in the model subspaces K Θ 2 =H 2 ΘH 2 of the Hardy class H2 in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels k λ n (z)=(1-Θ(λ n ) ¯Θ(z))/(z-λ ¯ n ) under “small” perturbations of the points λ n . We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces K Θ 2 and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.
DOI : https://doi.org/10.5802/aif.2165
Classification:  46E22,  42C15,  30D55,  47B32
Mots clés: fonction intérieure, espace modèle, noyaux reproduisant, base de Riesz, frame, stabilité
@article{AIF_2005__55_7_2399_0,
     author = {Baranov, Anton},
     title = {Stability of the bases and frames reproducing kernels in model spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     pages = {2399-2422},
     doi = {10.5802/aif.2165},
     zbl = {1101.30036},
     mrnumber = {2207388},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_7_2399_0}
}
Baranov, Anton. Stability of the bases and frames reproducing kernels in model spaces. Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2399-2422. doi : 10.5802/aif.2165. https://www.numdam.org/item/AIF_2005__55_7_2399_0/

[1] Ahern, P. R.; Clark, D. N. Radial limits and invariant subspaces, Amer. J. Math., Tome 92 (1970) no. 2, pp. 332-342 | Article | MR 262511 | Zbl 0197.39202

[2] Aleksandrov, A. B. Invariant subspaces of shift operators. An axiomatic approach, J. Soviet Math., Tome 22 (1983), pp. 1695-1708 (Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 7-26; English transl.) | Article | Zbl 0517.47019

[3] Aleksandrov, A. B. A simple proof of the Volberg-Treil theorem on the embedding of coinvariant subspaces of the shift operator, J. Math. Sci., Tome 5 (1997) no. 2, pp. 1773-1778 (Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 217 (1994), 26-35; English transl.) | Article | MR 1327512 | Zbl 0907.47001

[4] Aleksandrov, A. B. Embedding theorems for coinvariant subspaces of the shift operator. II, J. Math. Sci., Tome 110 (2002) no. 5, pp. 2907-2929 (Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 262 (1999), 5-48; English transl.) | Article | MR 1734326 | Zbl 1060.30043

[5] Baranov, A. D. The Bernstein inequality in the de Branges spaces and embedding theorems, Amer. Math. Soc., Ser. 2, Tome 209 (2003), pp. 21-49 (Proc. St. Petersburg Math. Soc., 9 (2001), 23-53; English transl.) | MR 2018371 | Zbl 1044.30012

[6] Baranov, A. D. Weighted Bernstein-type inequalities and embedding theorems for shift-coinvariant subspaces, Algebra i Analiz, Tome 15 (2003) no. 5, pp. 138-168 (English transl.: St. Petersburg Math. J., 15 (2004), 5, 733-752) | MR 2068792 | Zbl 1070.47021

[7] Baranov, A. D. Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings, J. Funct. Anal., Tome 223 (2005) no. 1, pp. 116-146 | Article | MR 2139883 | Zbl 02183823

[8] Boricheva, I. Geometric properties of projections of reproducing kernels on z * -invariant subspaces of H 2 , J. Funct. Anal., Tome 161 (1999) no. 2, pp. 397-417 | Article | MR 1674647 | Zbl 0939.30005

[9] Borwein, P.; Erdelyi, T. Sharp extensions of Bernstein's inequality to rational spaces, Mathematika, Tome 43 (1996) no. 2, pp. 413-423 | Article | MR 1433285 | Zbl 0869.41010

[10] Branges, L. De Hilbert spaces of entire functions, Prentice Hall, Englewood Cliffs (NJ) (1968) | MR 229011 | Zbl 0157.43301

[11] Clark, D. N. One-dimensional perturbations of restricted shifts, J. Anal. Math., Tome 25 (1972), pp. 169-191 | Article | MR 301534 | Zbl 0252.47010

[12] Cohn, W. S. Radial limits and star invariant subspaces of bounded mean oscillation, Amer. J. Math., Tome 108 (1986) no. 3, pp. 719-749 | Article | MR 844637 | Zbl 0607.30034

[13] Cohn, W. S. Carleson measures and operators on star-invariant subspaces, J. Oper. Theory, Tome 15 (1986) no. 1, pp. 181-202 | MR 816238 | Zbl 0615.47025

[14] Cohn, W. S. On fractional derivatives and star invariant subspaces, Michigan Math. J., Tome 34 (1987) no. 3, pp. 391-406 | Article | MR 911813 | Zbl 0629.30037

[15] Dyakonov, K. M. Entire functions of exponential type and model subspaces in H p , J. Math. Sci., Tome 71 (1994) no. 1, pp. 2222-2233 (Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 190 (1991), 81-100; English transl.) | Article | MR 1111913 | Zbl 0827.30020

[16] Dyakonov, K. M. Smooth functions in the range of a Hankel operator, Indiana Univ. Math. J., Tome 43 (1994), pp. 805-838 | Article | MR 1305948 | Zbl 0821.30026

[17] Dyakonov, K. M. Differentiation in star-invariant subspaces I, II, J. Funct. Anal., Tome 192 (2002) no. 2, pp. 364-409 | Article | MR 1923406 | Zbl 1011.47005

[18] Fricain, E. Bases of reproducing kernels in model spaces, J. Oper. Theory, Tome 46 (2001) no. 3 (suppl.), pp. 517-543 | MR 1897152 | Zbl 0995.46021

[19] Fricain, E. Complétude des noyaux reproduisants dans les espaces modèles, Ann. Inst. Fourier (Grenoble), Tome 52 (2002) no. 2, pp. 661-686 | Article | Numdam | MR 1906486 | Zbl 1032.46040

[20] Hruscev, S. V.; Nikolskii, N. K.; Pavlov, B. S. Unconditional bases of exponentials and of reproducing kernels, Lecture Notes in Math., Tome 864 (1981), pp. 214-335 | Article | MR 643384 | Zbl 0466.46018

[21] Kadec, M. I. The exact value of the Paley-Wiener constant, Sov. Math. Dokl., Tome 5 (1964), pp. 559-561 (Dokl. Akad. Nauk SSSR, 155 (1964), 1253-1254; English transl.) | MR 162088 | Zbl 0196.42602

[22] Levin, M. B. An estimate of the derivative of a meromorphic function on the boundary of domain, Sov. Math. Dokl., Tome 15 (1974) no. 3, pp. 831-834 | MR 352468 | Zbl 0299.30028

[23] Lyubarskii, Yu. I.; Seip, K. Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's (A p ) condition, Rev. Mat. Iberoamericana, Tome 13 (1997) no. 2, pp. 361-376 | MR 1617649 | Zbl 0918.42003

[24] Nikolski, N. K. Treatise on the shift operator, Springer-Verlag, Berlin-Heidelberg (1986) | MR 827223 | Zbl 0587.47036

[25] Nikolski, N. K. Operators, functions, and systems: an easy reading. Vol. 1, Hardy, Hankel, and Toeplitz, AMS, Providence, RI (Mathematical Surveys and Monographs) Tome 92 (2002) | MR 1864396 | Zbl 1007.47001

[26] Nikolski, N. K. Operators, functions, and systems: an easy reading. Vol. 2, Model operators and systems, AMS, Providence, RI (Mathematical Surveys and Monographs) Tome 93 (2002) | MR 1892647 | Zbl 1007.47002

[27] Ortega-Cerda, J.; Seip, K. Fourier frames, Ann. of Math., Tome 155 (2002) no. 3, pp. 789-806 ((2)) | Article | MR 1923965 | Zbl 1015.42023

[28] Seip, K. On the connection between exponential bases and certain related sequences in L 2 (-π,π), J. Funct. Anal., Tome 130 (1995) no. 1, pp. 131-160 | Article | MR 1331980 | Zbl 0872.46006

[29] Volberg, A. L.; Treil, S. R. Embedding theorems for invariant subspaces of the inverse shift operator, J. Soviet Math., Tome 42 (1988) no. 2, pp. 1562-1572 (Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 149 (1986), 38-51; English transl.) | Article | MR 849293 | Zbl 0654.30027

[30] Young, R. M. An Introduction to Nonharmonic Fourier Series, Academic Press, New-York (1980) | MR 591684 | Zbl 0493.42001