Skip to Main Content


AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution


Functions with Disconnected Spectrum: Sampling, Interpolation, Translates

About this Title

Alexander M. Olevskii, Tel Aviv University, Tel Aviv, Israel and Alexander Ulanovskii, Stavanger University, Stavanger, Norway

Publication: University Lecture Series
Publication Year: 2016; Volume 65
ISBNs: 978-1-4704-2889-1 (print); 978-1-4704-3216-4 (online)
DOI: https://doi.org/10.1090/ulect/065
MathSciNet review: MR3468930
MSC: Primary 42-02; Secondary 42A10, 94A12

PDF View full volume as PDF

Read more about this volume

View other years and volumes:

Table of Contents

PDF Download chapters as PDF

Front/Back Matter

Chapters

References [Enhancements On Off] (What's this?)

References
  • Ahlfors, Lars V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd ed, New York, NY, McGraw–Hill, 1979.
  • Amrein, W. O., Berthier, A.M. On support properties of $L^p$-functions and their Fourier transforms. J. Funct. Anal. 24 (1977), no. 3, 258–267.
  • Arutyunyan F. G. Representation of measurable funtions of several variables by multiple trigonometric series. Math. Sb. 126:2 (168) (1985), 267–285 (in Russian). English translation in Math. USSR. Sb. 54 (1986), 259–277.
  • Atzmon, A., Olevskii, A. Completeness of integer translates in function spaces on $\mathbb {R}$. J. Approx. Theory 87 (1996), no. 3, 291–327.
  • Au-Yeung, E., Benedetto, J. J. Generalized Fourier frames in terms of balayage. J. Fourier Anal. Appl. 21 (2015), no. 3, 472–508.
  • Avdonin, S. A. On the question of Riesz bases of exponential functions in $L^2$. Vestnik Leningrad. Univ. 13 (1974), 5–12 (in Russian). English translation in Vestnik Leningrad. Univ., Math. 7 (1979), 203–211.
  • Baiocchi, C., Komornik, V., Loreti, P. Théorèmes du type Ingham et application à la théorie du contrôle. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 4, 453–458.
  • Banaszczyk, W. Inequalities for convex bodies and polar reciprocal lattices in $\mathbb {R}^n$. Discrete Comput. Geom. 13 (1995), no. 2, 217–231.
  • Bari, N. K. Trigonometric Series, Fizmatgiz, Moscow, 1961 (in Russian). English translation: Bary, N. K. A Treatise on Trigonometric Series. Vols. I, II, The Macmillan Co., New York, 1964.
  • Baranov, A., Belov, Yu., Ulanovskii, A., Gap theorem for separated sequences without pain. To appear in J. Fourier Anal. Appl.
  • Batson J., Spielman D. A., Srivastava N. Twice-Ramanujan sparsifiers. SIAM Rev., 56(2) (2014), 315–334.
  • Benedicks, M. On Fourier transforms of functions supported on sets of finite Lebesgue measure. Royal Institute of Technology, Stockholm (1974), preprint. J. Math. Anal. Appl. 106, (1985), no. 1, 180–183.
  • Bernstein, S. N. Sur une propriete des fonctions entieres. C. R. 176 (1923).
  • Bernstein, S. N. Lecons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle. Gauthier–Villars, Paris, 1926.
  • Bernstein, S. N. The extension of properties of trigonometric polynomials to entire functions of finite degree. Izv. Akad. Nauk SSSR Ser. Math. 12 (1948), 421–444 (in Russian).
  • Beurling, A. and Malliavin, P. On Fourier transforms of measures with compact support. Acta. Math. 107 (1962), 291–309.
  • Beurling, A., Malliavin, P. On the closure of characters and the zeros of entire functions. Acta Math., 118 (1967), 79–93.
  • Beurling, A. Balayage of Fourier–Stiltjes transforms. In: The collected Works of Arne Beurling, v. 2, Harmonic Analysis. Birkhauser, Boston, 1989.
  • Beurling, A. Interpolation for an interval in $\mathbb {R}$. In: The Collected Works of Arne Beurling, v. 2, Harmonic Analysis. Birkhäuser, Boston, 1989.
  • Beurling, A. Local Harmonic analysis with some applications to differential operators. In: The collected Works of Arne Beurling, v. 2, Harmonic Analysis. Birkhauser, Boston, 1989.
  • Bezuglaya, L., Katsnelson, V. The sampling theorem for functions with limited multi-band spectrum. Z. Anal. Anwendungen 12 (1993), no. 3, 511–534.
  • Blank, N. Generating sets for Beurling algebras. J. Approx. Theory 140 (2006), no. 1, 61–70.
  • Blank, N., Ulanovskii, A. On Cartwright’s theorem. To appear in Proc. Amer. Math. Soc.
  • Boas, R. P., Jr. The Derivative of a trigonometric integral. J. London Math. Soc. s1-12, (1937), no. 3, 164–165.
  • Bourgain, J., Tzafriri, L. Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math. 57 (1987), no. 2, 137–224.
  • Bruna, J., Olevskii, A., Ulanovskii, A. Completeness in $L^1(\mathbb {R})$ of discrete translates. Rev. Mat. Iberoamerica 22 (2006), no. 1, 1–16.
  • Candes, E., Romberg, J., Tao, T. Robust uncertainty principle: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 (2006), 489–509.
  • Casazza, P. G., Tremain, J. The Kadison-Singer problem in mathematics and engineering. Proc. Natl. Acad. Sci. USA 103 (2006), no. 7, 2032–2039.
  • Cassels, J. W. S. An Introduction to the Geometry of Numbers, Springer-Verlag, 1971.
  • Clunie, J., Rahman, Q. I., Walker, W. J. On entire functions of exponential type bounded on the real axis. J. Lond. Math. Soc. (2) 61(1) (2000), 163–176.
  • Christensen, O. An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2002.
  • Christensen, O., Deng, B., Heil, C. Density of Gabor frames. Appl. Comput. Harmon. Anal., 7 (1999), 292-304.
  • Davtjan, R. S. The representation of measurable functions by Fourier integrals. Akad. Nauk Armjan. SSR Dokl. 53 (1971), 203–207 (in Russian).
  • Daubechies, I. Ten lectures on wavelets, CBMS–NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
  • Duffin, R. J., Schaeffer, A. C. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72 (1952), 341–366.
  • Faxen, B. On approximation by equidistant translates. Preprint (1996). Royal Inst. of Technology, Stockholm, Sweden.
  • Fuglede, B. Commuting self-adjoint partial differential operators and a group theoretic problem., J. Funct. Anal. 16 (1974), 101–121.
  • Fuglede, B. Orthogonal exponentials on the ball. Exposition. Math. 19 (2001), no. 3, 267–272.
  • Furstenberg, H., Katznelson, Y., Ornstein, D. S. The ergodic theoretical proof of Szemerédi’s theorem. Bull. Amer. Math. Soc. 7 (3) (1982), 527–552.
  • Gluskin, E., Olevskii, A. Invertibility of sub-matrices and the octahedron width theorem. Israel J. Math. 186 (2011), 61–68.
  • Grepstad, S., Lev, N. Universal sampling, quasicrystals and bounded remainder sets. C. R. Math. Acad. Sci. Paris 352 (2014), no. 7-8, 633–638.
  • Gröchenig, K., Razafinjatovo, H. On Landau’s necessary density conditions for sampling and interpolation of band–limited functions. J. London Math. Soc. (2) 54 (1996), no. 3, 557–565.
  • Han, D., Larson, D. R. Frames, Bases and Group Representations, Mem. Amer. Math. Soc. 147, 2000.
  • Hartman, P. The divergence of non-harmonic gap series. Duke Math. J. 9 (1942), 404–405.
  • Harvey, N. J. A., Olver, N. Pipage rounding, pessimistic estimators and matrix concentration. Proc. of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (2014), 926–945.
  • Heil, C. A basis theory primer. Expanded edition, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2011.
  • Helson, H. Harmonic Analysis, Addison–Wesley, Reading, MA, 1983.
  • Hewitt, E., Ross, K. A. Abstract harmonic analysis. Volume II, Springer-Verlag, Berlin, 1970.
  • Higgins, J. R. Sampling Theory in Fourier and Signal Analysis. Foundations, Clarendon Press. Oxford. 1996.
  • Hörmander, L. Some inequalities for functions of exponential type. Math. Scand. 3, (1955), 21–27.
  • Horn, R.A. and Johnson, C.R. Topics in matrix analysis, Cambridge Univ. Press, Cambridge, 1994.
  • Khrushchev, S. V., Nikolski, N. K., Pavlov, B. S. Unconditional bases of exponentials and of reproducing kernels. Complex analysis and spectral theory (Leningrad, 1979/1980), 214–335 (in Russian). English translation in Lecture Notes in Math., 864, Springer, Berlin–New York, 1981.
  • Ingham, A. E. Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936), 367–379.
  • Iosevich, A., Katz, N., Pedersen, S. Fourier bases and a distance problem of Erdős. Math. Res. Lett. 6 (1999), no. 2, 251–255.
  • Ivashov-Musatov, O. S. On coefficients of trigonometric null-series. Izv. Acad. Nauk SSSR, 21 (1957), 559–578 (in Russian).
  • Kadec, M. I. The exact value of the Paley-Wiener constant. Dokl. Akad. Nauk SSSR 155 (1964), 1253–1254 (in Russian).
  • Kahane, J.-P. Sur les fonctions moyenne-périodiques bornées. Ann. Inst. Fourier (Grenoble), 7 (1957), 293–314.
  • Kahane, J.-P. Pseudopériodicité et séries de Fourier lacunaries. Ann. Sci. Ecole Norm. Sup. 79 (1962), 93–150.
  • Kahane, J.-P. Turan’s method and compressive sampling. arXiv:1110.4482 (2011).
  • Kahane, J.-P. and Salem, R. Ensembles Parfaits et Series Trigonometriques. 2nd ed., Hermann, 1994.
  • Kashin, B. S. On some properties of matrices of bounded operators from the space $l^n_2$ into $l^m_2$. Izv. Akad. Nauk Armyan SSR Ser. Mat., 15 (1980), 379–394.
  • Katznelson, Y. An Introduction to Harmonic Analysis, Cambridge Universiy Press, 2004.
  • Katsnelson, V. E. Equivalent norms in spaces of entire functions. Mat. Sb. (N.S.) 92, no. 134 (1973), 34–54. (in Russian) English translation in Math. USSR-Sb. 21 (1973), 33–55.
  • Khavin, V. P., Mashreghi, J., Nazarov, F. Beurling-Malliavin multiplier theorem: the seventh proof. St. Petersburg Math. J. 17 (2006), 699–744.
  • Khavin, V. P., Jöricke, B. The uncertainty principle in harmonic analysis, Springer-Verlag, Berlin, 1994.
  • Khinchin, A. Three Pearls of Number Theory, Dover Publications, Minneola, NY, 1998.
  • Kohlenberg, A. Exact interpolation of band-limited functions. J. Appl. Phys. 24:12, (1953), 1432–1436.
  • Kolmogorov, A. N., Petrov, A. A. and Smirnov, Yu. M. A formula of Gauss in the theory of the method of least squares. Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), no. 6, 561–566. (in Russian).
  • Komornik, V., Loreti, P. Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.
  • Konyagin, S. V. Limits of indeterminacy of trigonometric series. (in Russian) Mat. Zametki 44 no. 6 (1988), 770–784, 862. English translation in Math. Notes 44 no. 5-6 (1989), 910–920.
  • Koosis, P. The Logarithmic Integral. II, Cambridge Studies in Advanced Mathematics, 21. Cambridge University Press, Cambridge, 1992.
  • Koosis, P. Introduction to $H^p$ Spaces, Second edition. Cambridge Tracts in Math. 115, Cambridge Univ. Press, Cambridge, 1998.
  • Kozma, G. and Lev, N. Exponential Riesz bases, discrepancy of irrational rotations and BMO. J. Fourier Anal. Appl. 17 (2011), no. 5, 879–898.
  • Kozma, G., Nitzan, S. Combining Riesz bases. Invent. Math. 199 (2015), no. 1, 267–285.
  • Kozma, G., Olevskii, A. Representation of non-periodic functions by trigonometric series with almost integer frequencies. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 4, 275–280.
  • Kozma, G., Olevskii, A. Menshov representation spectra. J. Anal. Math. 84 (2001), 361–393.
  • Kozma, G., Olevskii, A. Random Menshov spectra. Proc. Amer. Math. Soc. 131 no. 6 (2003), 1901–1906.
  • Kozma, G., Olevskii, A. Analytic representation of functions and a new quasi-analyticity threshold. Ann. of Math. (2) 164, no. 3 (2006), 1033–1064.
  • Körner, T. W. On the representation of functions by trigonometric series. 100 ans apr$\grave {e}$s. Th.-J. Stieltjes. Ann. Fac. Sci. Toulouse Math. 6, (1996), 77–119.
  • Laba, I. Fuglede’s conjecture for a union of two intervals. Proc. Amer. Math. Soc. 129 (2001), no. 10, 2965–2972.
  • Landau, H. J. A sparse regular sequence of exponentials closed on large sets. Bull. Amer. Math. Soc. 70 (1964), 566–569.
  • Landau, H. J. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117 (1967), 37–52.
  • Lawton, W. Minimal sequences and the Kadison-Singer problem. Bull. Malaysian Math. Soc. (2), 33 (2010), no. 2, 169–176.
  • Lev, N., Olevskii, A. Wiener’s “closure of translates” problem and Piatetski–Shapiro’s uniqueness phenomenon. Ann. of Math. (2) 174, (2011), 519-541.
  • Levin, B. Ya. Lectures on entire functions. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko. Translated from the Russian manuscript by Tkachenko. Translations of Mathematical Monographs, 150. American Mathematical Society, Providence, RI, 1996.
  • Lin, V. Ja. On equivalent norms in the space of square integrable entire functions of exponential type. Mat. Sb. (N.S.) 67 (1965), no. 109, 586–608 (in Russian).
  • Liu, B., Liu, R. Upper Beurling density of systems formed by translates of finite sets of elements in $L^p(\mathbb {R}^d)$. Banach J. Math. Anal. 6 (2012), no. 2, 86–97.
  • Logvinenko, V.N., Sereda, Yu.F. Equivalent norms in spaces of entire functions of exponential type. Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. (Khar’kov), 20 (1974), 102–111 (in Russian).
  • Lunin, A. A. On operator norms of submatrices. Mat. Zametki 45 (1989), no. 3, 94–100, 128 (in Russian).
  • Lyubarskii, Yu., Seip, K. Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets. J. Fourier Anal. Appl. 3 (1997), no. 5, 597–615.
  • Lyubarskii, Yu., Rashkovskii, A. Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons. Ark. Mat. 38 (2000), no. 1, 139–170.
  • Makarov, N., Poltoratski, A. Beurling-Malliavin theory for Toeplitz kernels. Invent. Math. 180 (2010), no. 3, 443–480.
  • Marcus, A., Spielman, D. A., Srivastava, N. Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem. Ann. of Math. (2) 182 (2015), no. 1, 327–350.
  • Matei, B., Meyer, Y. A variant of compressed sensing. Rev. Mat. Iberoamerica 25 (2009), no. 2, 669–692.
  • Matei, B., Meyer, Y. Simple quasicrystals are sets of stable sampling. Complex Var. Elliptic Eq. 55 (2010), no. 8-10, 947–964.
  • Meyer, Y. Algebraic numbers and harmonic analysis. North–Holland Mathematical Library, Vol. 2. North–Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1972.
  • Meyer, Y. Wavelets and Operators., Translated from the 1990 French original by D. H. Salinger. Cambridge Studies in Advanced Mathematics, 37, Cambridge University Press, Cambridge, 1992.
  • Meyer, Y. Quasicrystalls, diophantine approximations and algebraic numbers. In: Beyond quasicrystalls (Les Houches, 1994), 3-16. Springer, Berlin, 1995.
  • Mitkovski, M., Poltoratski, A. Pólya sequences, Toeplitz kernels and gap theorems. Adv. Math. 224 (2010), no. 3, 1057–1070.
  • Nazarov, F. L. Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz 5, no. 4 (1993), 3–66. (in Russian). English translation in St. Petersburg Math. J. 5 (1994), no. 4, 663–717.
  • Nazarov, F. L. The Hörmander proof of the Bourgain–Milman theorem. Geometric aspects of functional analysis, Lecture Notes in Math., 2050, Springer, Heidelberg, (2012), 335–343.
  • Nikolskii, N. Selected Problems in Weighted Approximation and Spectral Analysis, Proceedings of the Steklov Institute of Mathematics 120, 1974 (in Russian). English translation in Amer. Math. Soc., Providence, RI, 1976.
  • Nitzan, S. Private communication (2016).
  • Nitzan, S., Olevskii, A. Sparse exponential systems: completeness with estimates. Israel J. Math. 158 (2007), 205–215.
  • Nitzan, S., Olevskii, A. Quasi-frames of translates. C. R. Acad. Sci. Paris, Sér. I Math. 347 (2009), 739–742.
  • Nitzan, S., Olevskii, A. Revisiting Landau’s density theorems for Paley–Wiener spaces. C. R. Acad. Sci. Paris, Sér. I Math. 350 (2012), no. 9–10, 509–512.
  • Nitzan, S., Olevskii, A., Ulanovskii, A. A few remarks on sampling of signals with small spectrum. Proc. Steklov Inst. Math. 280 (2013), 240–247.
  • Nitzan, S., Olevskii, A., Ulanovskii, A. Exponential frames for unbounded sets. Proc. Amer. Math. Soc. 144 (2016), 109–118.
  • Olevskii, A. Fourier Series with Respect to General Orthogonal Systems, Translated from the Russian by B. P. Marshall and H. J. Christoffers. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 86. Springer-Verlag, New York-Heidelberg, 1975.
  • Olevskii, A. Modification of functions and Fourier series. Uspekhi Mat. Nauk 40 (1985), 157–193 (in Russian). English translation in Russian Math. Surveys 40 (1985), 181–224.
  • Olevskii, A., Completeness in $L^2(\mathbb {R})$ of almost integer translates. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 9, 987–991.
  • Olevskii, A. Representation of functions by exponentials with positive frequencies. Uspekhi Mat. Nauk 59 (2004), 169–178 (in Russian). English translation in Russian Math. Surveys 59 (2004), 171–180.
  • Olevskii, A. Sampling, interpolation, translates. In Euporean Congress of Math. Krakow 2012 (2014), 489–501.
  • Olevskii, V. On orthonormal bases and translates, J. Approx. Theory, 202 (2016), 1–4.
  • Olevskii, A., Ulanovskii, A. Almost integer translates. Do nice generators exist? J. Fourier Anal. Appl. 10 (2004), no. 1, 93–104.
  • Olevskii, A., Ulanovskii, A. Universal sampling of band-limited signals. C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 927–931.
  • Olevskii, A., Ulanovskii, A. Universal sampling and interpolation of band-limited signals. Geom. Funct. Anal. 18 (2008), 1029–1052.
  • Olevskii, A., Ulanovskii, A. Interpolation by functions with small spectra. C. R. Math. Acad. Sci. Paris 345 (2007), no. 5, 261–264.
  • Olevskii, A., Ulanovskii, A. Interpolation in Bernstein and Paley–Wiener spaces. J. Funct. Anal. 256 (2009), 3257–3278.
  • Olevskii, A., Ulanovskii, A. Approximation of discrete functions and size of spectrum. St. Petersburg Math. J. 21 (2010), 1015–1025.
  • Olevskii, A., Ulanovskii, A. On Ingham-type interpolation in $\mathbb {R}^n$. C. R. Math. Acad. Sci. Paris, 348 (2010), 807–810.
  • Olevskii, A., Ulanovskii, A. Uniqueness sets for unbounded spectra. C. R. Math. Acad. Sci. Paris, 349 (2011), 679–681.
  • Olevskii, A., Ulanovskii, A. On multi-dimensional sampling and interpolation. Anal. Math. Phys. 2 (2012), no. 2, 149–170.
  • Olevskii, A., Ulanovskii, A. Discrete translates in $L^p(\mathbb {R})$. Oberwolfach Preprints (2015), OWP2015-09.
  • Olevskii, A., Ulanovskii, A. On irregular sampling in Bernstein spaces. C. R. Math. Acad. Sci. Paris, 353 (2015), no. 1, 47–50.
  • Olevskii, A., Ulanovskii, A. On the duality between sampling and interpolation, Anal. Math. 42 (2016), no. 1, 43–53.
  • Olson, T. E., Zalik, R. A. Nonexistence of a Riesz basis of translates. In “Approximation Theory”. Lecture Notes in Pure and Applied Math. 138, Dekker, New York (1992), 401–408.
  • Ortega-Cerdá, J., Seip, K. Fourier frames. Ann. of Math. (2) 155 (2002), no. 3, 789–806.
  • Osgood, B., Siripuram, A., Wu, W. Discrete sampling and interpolation: universal sampling sets for discrete bandlimited spaces. IEEE Trans. Inform. Theory, 58, (2012), no. 7, 4176–4200.
  • Paley, R., Wiener, N. Fourier Transform in the Complex Domain, American Mathematical Society, Providence, RI, 1934.
  • Panejah, B. P. Some theorems of Paley–Wiener type. Dokl. Akad. Nauk SSSR 138 (1961), 47–50 (in Russian). English translation in Soviet Math. Dokl. 2, (1961), 533–536.
  • Pavlov, B. S. Basicity of an exponential systems and Muckenhoupt’s condition. Soviet Math. Dokl. 20 (1979), 655–659.
  • Privalov, I. I. Boundary Properties of Analytic Functions, Second edition, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1950 (in Russian).
  • Ramanathan, J., Steger, T. Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2 (1995), 148–153.
  • Redheffer, R. M., Young, R. M. Completeness and basis properties of complex exponentials. Trans. Amer. Math. Soc. 277 (1983), no. 1, 93–111.
  • Ricaud, B., Torrésani, B. A survey of uncertainty principles and some signal processing applications. Adv. Comp. Math., 40 (2014), no. 3, 629–650.
  • Rogers, C. A. Packing and Covering, Cambridge University Press, Cambridge, 2008.
  • Roth, K. F. On certain sets of integers. J. London Math. Soc. 28 (1953), 104-109.
  • Rudin, W. Functional Analysis, McGraw-Hill, New York, 1991.
  • Sedletskii, A. M. Approximation by shifts and completeness of weighted systems of exponentials in $L^2(\mathbb {R})$. Math. USSR Sb. 51 (1985), no. 1, 92–107.
  • Seip, K. A simple construction of exponential bases in $L^2$ of the union of several intervals. Proc. Edinburgh Math. Soc. (2) 38:1 (1995), 171–177.
  • Seip, K. Interpolation and Sampling in Spaces of Analytic Functions, University Lecture Series, 33. American Mathematical Society, Providence, RI, 2004.
  • Shapiro, G. S. Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies. Trans. Amer. Math. Soc. 225 (1977), 183–198.
  • Szemerédi, E. On sets of integers containing no $k$ elements in arithmetic progression. Acta Arith. 27 (1975), 199–245.
  • Ulanovskii, A. On Landau’s phenomenon in $\mathbb {R}^n$. Math. Scand. 88 (2001), no. 1, 72–78.
  • Wiener N. Tauberian theorems. Annals of Math. 33 (1932), 1-100.
  • Wiener N. The Fourier Integral and Certain of its Applications, Re-issued with Foreword in the Cambridge Mathematical Library series, 1988. Cambridge University Press, 1933.
  • Weaver, N. The Kadison-Singer problem in discrepancy theory. Discrete Math. 278 (2004), 227–239.
  • Young, R.M. An introduction to Nonharmonic Fourier Series, Academic Press. 2001.
  • Zygmund, A. Trigonometric series. Vol. I, II. Third edition, With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. xii; Vol. I, II.