Functional anaysis, Harmonic analysis
A Rudin–de Leeuw type theorem for functions with spectral gaps
Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 797-803.

Our starting point is a theorem of de Leeuw and Rudin that describes the extreme points of the unit ball in the Hardy space ${H}^{1}$. We extend this result to subspaces of ${H}^{1}$ formed by functions with smaller spectra. More precisely, given a finite set $𝒦$ of positive integers, we prove a Rudin–de Leeuw type theorem for the unit ball of ${H}_{𝒦}^{1}$, the space of functions $f\in {H}^{1}$ whose Fourier coefficients $\stackrel{^}{f}\left(k\right)$ vanish for all $k\in 𝒦$.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.208
Classification: 30H10,  30J10,  42A32,  46A55
Konstantin M. Dyakonov 1, 2

1 Departament de Matemàtiques i Informàtica, IMUB, BGSMath, Universitat de Barcelona, Gran Via 585, E-08007 Barcelona, Spain
2 ICREA, Pg. Lluís Companys 23, E-08010 Barcelona, Spain
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Konstantin M. Dyakonov. A Rudin–de Leeuw type theorem for functions with spectral gaps. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 797-803. doi : 10.5802/crmath.208. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.208/

[1] S. Axler Linear algebra done right, Undergraduate Texts in Mathematics, Springer, 2015 | Zbl

[2] Ronald G. Douglas; Harold S. Shapiro; Allen L. Shields Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier, Volume 20 (1970) no. 1, pp. 37-76 | DOI | MR | Zbl

[3] Konstantin M. Dyakonov The geometry of the unit ball in the space ${K}_{\theta }^{1}$, Geometric problems of the theory of functions and sets, Kalinin. Gos. Univ., Kalinin, 1987, pp. 52-54 | Zbl

[4] Konstantin M. Dyakonov Moduli and arguments of analytic functions from subspaces in ${H}^{p}$ that are invariant under the backward shift operator, Sib. Math. J., Volume 31 (1990) no. 6, pp. 926-939 translation from Sib. Mat. Zh. 31, No. 6 (1990), 64–79 | MR | Zbl

[5] Konstantin M. Dyakonov Interpolating functions of minimal norm, star-invariant subspaces, and kernels of Toeplitz operators, Proc. Am. Math. Soc., Volume 116 (1992) no. 4, pp. 1007-1013 | MR | Zbl

[6] Konstantin M. Dyakonov Polynomials and entire functions: zeros and geometry of the unit ball, Math. Res. Lett., Volume 7 (2000) no. 4, pp. 393-404 | DOI | MR | Zbl

[7] Konstantin M. Dyakonov Lacunary polynomials in ${L}^{1}$: geometry of the unit sphere, Adv. Math., Volume 381 (2021), 107607, 24 pages | MR | Zbl

[8] Konstantin M. Dyakonov Nearly outer functions as extreme points in punctured Hardy spaces (2021) (https://arxiv.org/abs/2102.05857)

[9] Theodore W. Gamelin Uniform algebras, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1969 | MR | Zbl

[10] John B. Garnett Bounded analytic functions, Graduate Texts in Mathematics, 236, Springer, 2007 | MR | Zbl

[11] Kenneth Hoffman Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962 | Zbl

[12] Paul Koosis Introduction to ${H}_{p}$ spaces, Cambridge Tracts in Mathematics, 115, Cambridge University Press, 1998 (with two appendices by V. P. Havin) | MR | Zbl

[13] Karel de Leeuw; W. Rudin Extreme points and extremum problems in ${H}_{1}$, Pac. J. Math., Volume 8 (1958), pp. 467-485 | DOI | MR | Zbl

[14] Karel de Leeuw; Walter Rudin; John Wermer The isometries of some function spaces, Proc. Am. Math. Soc., Volume 11 (1960), pp. 694-698 | DOI | MR | Zbl

[15] Nikolaĭ K. Nikolski Operators, Functions, and Systems: An Easy Reading. Volume 2: Model operators and systems, Mathematical Surveys and Monographs, 93, American Mathematical Society, 2002 | MR | Zbl

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