##
**Spectral gaps for sets and measures.**
*(English)*
Zbl 1245.42008

The uncertainty principle in harmonic analysis has motivated much research in harmonic analysis. The principle roughly states that it is not possible for both a function (or measure) and its Fourier transform to be small, unless it is identically zero. There are many ways to formulate problems arising from this principle by considering different interpretations of ‘small’. In this paper, smallness is understood in terms of the sparseness of the supports of the measure and its transform, the so-called gap problem. The objective of the gap problem is to obtain quantitative estimates relating the size of gaps in the support of \(\widehat{\mu }\) with the size of the support of \(\mu \). A classical theorem of this type is Beurling’s gap theorem which says that if the sequence of gaps in the support of \(\mu \) is ‘long’, then the support of \(\widehat{\mu }\) cannot have any gaps, unless \(\mu =0\).

Given \(X\) a closed subset of \(\mathbb{R}\), denote by \(G_{X}\) the supremum of the size of the gap in the support of \(\widehat{\mu }\), taken over all finite measures \(\mu \) supported on \(X\). The author introduces a characteristic of \(X\), called \(C_{X}\), which incorporates both a density and an energy condition. The main result is the proof that \(G_{X}=2\pi C_{X}\). The proof uses Toeplitz operators and other important ideas from analytic function theory, such as the Beurling-Malliavin multiplier theorem.

The paper is long and technical, but the ideas are well motivated, both mathematically and physically. The author also gives various formulations and discusses related problems such as de Brange’s result on the existence of a measure with a given spectral gap, the problem of uniform convergence of continuous functions by trigonometric polynomials, and the type problem, the problem of finding the least \(a\) such that the exponential functions \(e^{i\lambda x}\), with \(\lambda \in [0,a]\), span \(L^{2}(\mu )\). He concludes with a Toeplitz version of de Branges’ theorem.

Given \(X\) a closed subset of \(\mathbb{R}\), denote by \(G_{X}\) the supremum of the size of the gap in the support of \(\widehat{\mu }\), taken over all finite measures \(\mu \) supported on \(X\). The author introduces a characteristic of \(X\), called \(C_{X}\), which incorporates both a density and an energy condition. The main result is the proof that \(G_{X}=2\pi C_{X}\). The proof uses Toeplitz operators and other important ideas from analytic function theory, such as the Beurling-Malliavin multiplier theorem.

The paper is long and technical, but the ideas are well motivated, both mathematically and physically. The author also gives various formulations and discusses related problems such as de Brange’s result on the existence of a measure with a given spectral gap, the problem of uniform convergence of continuous functions by trigonometric polynomials, and the type problem, the problem of finding the least \(a\) such that the exponential functions \(e^{i\lambda x}\), with \(\lambda \in [0,a]\), span \(L^{2}(\mu )\). He concludes with a Toeplitz version of de Branges’ theorem.

Reviewer: Kathryn Hare (Waterloo)

### MSC:

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

43A05 | Measures on groups and semigroups, etc. |

47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |

### Keywords:

uncertainty principle in harmonic analysis; spectral gaps; Toeplitz operators; Beurling-Malliavin theory### References:

[1] | Aleksandrov, A. B., Isometric embeddings of co-invariant subspaces of the shift operator. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 232 (1996), 5–15, 213 (Russian); English translation in J. Math. Sci. (N. Y.), 92 (1998), 3543–3549. · Zbl 0907.30037 |

[2] | Benedicks, M., The support of functions and distributions with a spectral gap. Math. Scand., 55 (1984), 285–309. · Zbl 0577.42008 |

[3] | Beurling, A., On Quasianalyticity and General Distributions. Summer Institute, Stanford University, Stanford, CA, 1961. |

[4] | Beurling, A. & Malliavin, P., On Fourier transforms of measures with compact support. Acta Math., 107 (1962), 291–309. · Zbl 0127.32601 |

[5] | – On the closure of characters and the zeros of entire functions. Acta Math., 118 (1967), 79–93. · Zbl 0171.11901 |

[6] | Borichev, A. & Sodin, M., Weighted exponential approximation and non-classical orthogonal spectral measures. Adv. Math., 226 (2011), 2503–2545. · Zbl 1229.41018 |

[7] | de Branges, L., Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs, NJ, 1968. · Zbl 0157.43301 |

[8] | Coifman, R. R. & Weiss, G., Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 83 (1977), 569–645. · Zbl 0358.30023 |

[9] | Cornu, F. & Jancovici, B., On the two-dimensional Coulomb gas. J. Stat. Phys., 49 (1987), 33–56. · Zbl 0960.82502 |

[10] | Dym, H., On the span of trigonometric sums in weighted l 2 spaces, in Linear and Complex Analysis Problem Book 3, Part II, Lecture Notes in Math., 1574, pp. 87–88. Springer, Berlin–Heidelberg, 1994. |

[11] | Dym, H. & Mckean, H. P., Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Probability and Mathematical Statistics, 31. Academic Press, New York, 1976. · Zbl 0327.60029 |

[12] | Gel’fand, I. M. & Levitan, B. M., On the determination of a differential equation from its spectral function. Izv. Akad. Nauk SSSR Ser. Mat., 15 (1951), 309–360 (Russian); English translation in Amer. Math. Soc. Transl., 1 (1955), 253–304. |

[13] | Havin, V. & Jöricke, B., The Uncertainty Principle in Harmonic Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, 28. Springer, Berlin–Heidelberg, 1994. · Zbl 0827.42001 |

[14] | Kerov, S. V., Equilibrium and orthogonal polynomials. Algebra i Analiz, 12:6 (2000), 224-237 (Russian); English translation in St. Petersburg Math. J., 12 (2001), 1049–1059. |

[15] | Koosis, P., The Logarithmic Integral. I. Cambridge Studies in Advanced Mathematics, 12. Cambridge University Press, Cambridge, 1988. · Zbl 0665.30038 |

[16] | Kreĭn, M. G., On a problem of extrapolation of A. N. Kolmogoroff. Dokl. Acad. Nauk SSSR, 46 (1945), 306–309 (Russian). · Zbl 0063.03356 |

[17] | – On the transfer function of a one-dimensional boundary problem of the second order. Dokl. Akad. Nauk SSSR, 88 (1953), 405–408 (Russian). |

[18] | – On a basic approximation problem of the theory of extrapolation and filtration of stationary random processes. Dokl. Akad. Nauk SSSR, 94 (1954), 13–16 (Russian). · Zbl 0057.35002 |

[19] | Levinson, N., Gap and Density Theorems. American Mathematical Society Colloquium Publications, 26. Amer. Math. Soc., New York, 1940. · JFM 66.0332.01 |

[20] | Makarov, N. & Poltoratski, A., Meromorphic inner functions, Toeplitz kernels and the uncertainty principle, in Perspectives in Analysis, Math. Phys. Stud., 27, pp. 185–252. Springer, Berlin–Heidelberg, 2005. · Zbl 1118.47020 |

[21] | – Beurling–Malliavin theory for Toeplitz kernels. Invent. Math., 180 (2010), 443–480. · Zbl 1186.47025 |

[22] | Mitkovski, M. & Poltoratski, A., Pólya sequences, Toeplitz kernels and gap theorems. Adv. Math., 224 (2010), 1057–1070. · Zbl 1204.30018 |

[23] | Nienhuis, B., Coulomb gas formulation of two-dimensional phase transitions, in Phase Transitions and Critical Phenomena, Vol. 11, pp. 1–53. Academic Press, London, 1987. |

[24] | Nikol ’ skii, N. K., Bases of exponentials and values of reproducing kernels. Dokl. Akad. Nauk SSSR, 252 (1980), 1316–1320 (Russian); English translation in Soviet Math. Dokl., 21 (1980), 937–941. |

[25] | – Treatise on the Shift Operator. Grundlehren der Mathematischen Wissenschaften, 273. Springer, Berlin–Heidelberg, 1986. |

[26] | Poltoratski, A., On the boundary behavior of pseudocontinuable functions. Algebra i Analiz, 5:2 (1993), 189–210 (Russian); English translation in St. Petersburg Math. J., 5 (1994), 389–406. |

[27] | – Kreĭn’s spectral shift and perturbations of spectra of rank one. Algebra i Analiz, 10:5 (1998), 143–183 (Russian); English translation in St. Petersburg Math. J., 10 (1999), 833–859. |

[28] | – A problem on completeness of exponentials. Preprint, 2010. arXiv:1006.1840 [math.CA]. |

[29] | Poltoratski, A. & Sarason, D., Aleksandrov–Clark measures, in Recent Advances in Operator-Related Function Theory, Contemp. Math., 393, pp. 1–14. Amer. Math. Soc., Providence, RI, 2006. · Zbl 1102.30032 |

[30] | Šamaj, L., The statistical mechanics of the classical two-dimensional Coulomb gas is exactly solved. J. Phys. A, 36 (2003), 5913–5920. · Zbl 1041.82002 |

[31] | Simon, B., Spectral analysis of rank one perturbations and applications, in Mathematical Quantum Theory. II. Schrödinger Operators (Vancouver, BC, 1993), CRM Proc. Lecture Notes, 8, pp. 109–149. Amer. Math. Soc., Providence, RI, 1995. · Zbl 0824.47019 |

[32] | Sodin, M. & Yuditskii, P., Another approach to de Branges’ theorem on weighted polynomial approximation, in Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), Israel Math. Conf. Proc., 11, pp. 221–227. Bar-Ilan University, Ramat Gan, 1997. · Zbl 0897.41005 |

[33] | Vinogradov, S. A., Properties of multipliers of integrals of Cauchy-Stieltjes type, and some problems of factorization of analytic functions, in Mathematical Programming and Related Questions (Proc. Seventh Winter School, Drogobych, 1974), pp. 5–39 (Russian). Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.