My Holdings   Activate Remote Access

The Mathematics of Superoscillations

About this Title

Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa and Jeff Tollaksen

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1174
ISBNs: 978-1-4704-2324-7 (print); 978-1-4704-3709-1 (online)
Published electronically: February 1, 2017
Keywords:Superoscillations, convolution operators, infinite order differential operators, entire functions, Fourier transforms

View full volume PDF

View other years and numbers:

Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Physical motivations
  • Chapter 3. Basic mathematical properties of superoscillating sequences
  • Chapter 4. Function spaces of holomorphic functions with growth
  • Chapter 5. Schrödinger equation and superoscillations
  • Chapter 6. Superoscillating functions and convolution equations
  • Chapter 7. Superoscillating functions and operators
  • Chapter 8. Superoscillations in $SO(3)$


In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.

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

  • [1] Y. Aharonov, D. Albert, L. Vaidman, How the Results of a Measurement of a component of a spin particle can turn out to be ?, Phys. Rev. Lett. 60 (1988), 1351âĂŞ1354.
  • [2] Y. Aharonov, E. Ben-Reuven, S. Popescu, and D. Rohrlich, Perturbative induction of vector potentials, Phys. Rev. Lett. 65 (1990), no. 25, 3065–3067. MR 1082112,
  • [3] Yakir Aharonov, Peter G. Bergmann, and Joel L. Lebowitz, Time symmetry in the quantum process of measurement, Phys. Rev. (2) 134 (1964), B1410–B1416. MR 0163614
  • [4] Y. Aharonov and D. Bohm, Time in the quantum theory and the uncertainty relation for time and energy, Phys. Rev. (2) 122 (1961), 1649–1658. MR 0121086
  • [5] Y. Aharonov, A. Botero, Quantum averages of weak values, Physical Review A, 72 (2005), Art. No. 052111.
  • [6] Yakir Aharonov, Alonso Botero, Sandu Popescu, Benni Reznik, and Jeff Tollaksen, Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values, Phys. Lett. A 301 (2002), no. 3-4, 130–138. MR 1927989,
  • [7] Y. Aharonov, F. Colombo, S. Nussinov, I. Sabadini, D. C. Struppa, and J. Tollaksen, Superoscillation phenomena in $\rm SO(3)$, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), no. 2147, 3587–3600. MR 2988277,
  • [8] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, Some mathematical properties of superoscillations, J. Phys. A 44 (2011), no. 36, 365304, 16. MR 2826550,
  • [9] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, On some operators associated to superoscillations, Complex Anal. Oper. Theory 7 (2013), no. 4, 1299–1310. MR 3079857,
  • [10] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, On the Cauchy problem for the Schrödinger equation with superoscillatory initial data, J. Math. Pures Appl. (9) 99 (2013), no. 2, 165–173. MR 3007842,
  • [11] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, On superoscillations longevity: a windowed Fourier transform approach, in D. Struppa, J. Tollaksen (eds) Quantum Theory: A Two-Time Success Story, Springer 2013, 313âĂŞ325.
  • [12] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, Evolution of superoscillatory data, J. Phys. A 47 (2014), no. 20, 205301, 18. MR 3205922,
  • [13] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, Superoscillating sequences as solutions of generalized Schrödinger equations, J. Math. Pures Appl. (9) 103 (2015), no. 2, 522–534 (English, with English and French summaries). MR 3298368,
  • [14] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, Evolution of superoscillatory initial data in several variables in uniform electric field, Preprint 2015.
  • [15] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, Superoscillating sequences in several variables, J. Fourier Anal. Appl. 22 (2016), no. 4, 751–767. MR 3528397,
  • [16] Y. Aharonov, L. Davidovich, N. Zagury, Quantum Random Walks, Phys. Rev., A48 (1993), 1687.
  • [17] Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, L. Vaidman, Adiabatic Measurements on Metastable Systems, Phys. Rev. Lett., 77, (1996), p. 983.
  • [18] Y. Aharonov, S. Popescu, J. Tollaksen, A time-symmetric formulation of quantum mechanics, Physics Today, November 2010, p. 27.
  • [19] Y. Aharonov, S. Popescu, J. Tollaksen, Each moment of time is a new universe, in D. Struppa, J. Tollaksen (Eds), Quantum Theory: a Two-Time Success Story. Yakir Aharonov Festschrift. Springer, Milan, 2013.
  • [20] Yakir Aharonov and Daniel Rohrlich, Quantum paradoxes, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005. Quantum theory for the perplexed. MR 2327477
  • [21] Y. Aharonov, J. Tollaksen, New insights on Time-Symmetry in Quantum Mechanics, in Visions of Discovery: New Light on Physics, Cosmology And Consciousness, ed. R. Y. Chiao, M. L. Cohen, A. J. Leggett, W. D. Phillips, and C. L. Harper, Jr. Cambridge: Cambridge University Press, 2010.
  • [22] Yakir Aharonov and Lev Vaidman, Properties of a quantum system during the time interval between two measurements, Phys. Rev. A (3) 41 (1990), no. 1, 11–20. MR 1033952,
  • [23] Yakir Aharonov and Lev Vaidman, Complete description of a quantum system at a given time, J. Phys. A 24 (1991), no. 10, 2315–2328. MR 1118534
  • [24] Y. Aharonov, L. Vaidman, The Two-State Vector Formalism of Quantum Mechanics: an Updated Review, in Time in Quantum Mechanics, J. Muga, R. Sala Mayato and I. Egusquiza (Eds), Springer, Berlin, 2002.
  • [25] S. E. Ahnert and M. C. Payne, Linear optics implementation of weak values in Hardy’s paradox, Phys. Rev. A (3) 70 (2004), no. 4, 042102, 4. MR 2110118,
  • [26] T. Aoki, F. Colombo, I. Sabadini, D. C. Struppa, Continuity theorems for a class of convolution operators and applications to superoscillations, Preprint 2017.
  • [27] V. Balser, Summability of power series that are formal solutions of partial differential equations with constant coefficients, Sovrem. Mat. Fundam. Napravl. 1 (2003), 5–17 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 124 (2004), no. 4, 5085–5097. MR 2129124,
  • [28] Carlos A. Berenstein and Milos A. Dostal, Analytically uniform spaces and their applications to convolution equations, Lecture Notes in Mathematics, Vol. 256, Springer-Verlag, Berlin-New York, 1972. MR 0493316
  • [29] Carlos A. Berenstein and Daniele C. Struppa, Dirichlet series and convolution equations, Publ. Res. Inst. Math. Sci. 24 (1988), no. 5, 783–810. MR 985279,
  • [30] K. Berensteĭn and D. Struppa, Complex analysis and convolution equations, Current problems in mathematics. Fundamental directions, Vol. 54 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 5–111 (Russian). MR 1039621
  • [31] C. A. Berenstein and D. C. Struppa, Interpolation and Dirichlet series: a new approach, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989) Sem. Conf., vol. 8, EditEl, Rende, 1991, pp. 33–45. MR 1222205
  • [32] Carlos A. Berenstein and Daniele C. Struppa, Sheaves of holomorphic functions with growth conditions, $D$-modules and microlocal geometry (Lisbon, 1990) de Gruyter, Berlin, 1993, pp. 63–74. MR 1206013
  • [33] Carlos A. Berenstein and B. A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math. 33 (1979), no. 2, 109–143. MR 544846,
  • [34] C. A. Berenstein and B. A. Taylor, Interpolation problems in ${\bf C}^{n}$ with applications to harmonic analysis, J. Analyse Math. 38 (1980), 188–254. MR 600786
  • [35] G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, M. J. Padgett, Efficient Sorting of Orbital Angular Momentum States of Light, Physical Review Letters, 105 (2010), 153601.
  • [36] M. V. Berry, Faster than Fourier, in Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov, J.S. Anandan and J.L. Safko eds., World Scientific, Singapore, 55âĂŞ65, 1994.
  • [37] M. V. Berry, Evanescent and real waves in quantum billiards and Gaussian beams, J. Phys. A 27 (1994), no. 11, L391–L398. MR 1282572
  • [38] M. V. Berry, Exact nonparaxial transmission of subwavelength detail using superoscillations, J. Phys. A 46 (2013), no. 20, 205203, 15. MR 3055364,
  • [39] M. V. Berry, Representing superoscillations and narrow Gaussians with elementary functions, Milan J. Math. 84 (2016), no. 2, 217–230. MR 3574594,
  • [62] M. Berry, M. R. Dennis, Superoscillation in speckle patterns, Journal of Physics A: Mathematical and General, (2009).
  • [41] M. V. Berry and M. R. Dennis, Natural superoscillations in monochromatic waves in $D$ dimensions, J. Phys. A 42 (2009), no. 2, 022003, 8. MR 2525282,
  • [42] M. V. Berry, M. R. Dennis, B. McRoberts, and P. Shukla, Weak value distributions for spin 1/2, J. Phys. A 44 (2011), no. 20, 205301, 8. MR 2792436,
  • [43] M. V. Berry and S. Popescu, Evolution of quantum superoscillations and optical superresolution without evanescent waves, J. Phys. A 39 (2006), no. 22, 6965–6977. MR 2233265,
  • [44] M. V. Berry and Pragya Shukla, Pointer supershifts and superoscillations in weak measurements, J. Phys. A 45 (2012), no. 1, 015301, 14. MR 2871389,
  • [45] Y. K. Bliokh, Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect, Phys. Rev. Lett. 97 (4), Article Number: 043901, (2006).
  • [46] Y. K. Bliokh, I. V. Shadrivov, Y. S. Kivshar, Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams, Optics Letters 34 (3) (2009), 389âĂŞ391.
  • [47] F. E. Bond, C. R. Cahn, On sampling the zeros of bandwidth limited signals, IRE Trans. Infor. Theory 4 (3) (1958), 110âĂŞ113.
  • [48] Alonso Botero, Sampling weak values: A nonlinear Bayesian model for nonideal quantum measurements, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–The University of Texas at Austin. MR 2700297
  • [49] R. Brout, S. Massar, R. Parentani, S. Popescu, Ph. Spindel, Quantum back reaction on a classical field, Phys. Rev. D 52 (1995), 1119.
  • [50] N. Brunner, V. Scarani, M. Wegmüller, M. Legré, N. Gisin, Direct Measurement of Superluminal Group Velocity and Signal Velocity in an Optical Fiber, Phys. Rev. Lett. 93, 203902 (2004).
  • [51] N. Brunner, C. Simon, Measuring small longitudinal phase shifts: weak measurements or standard interferometry?, Phys. Rev. Lett. 105, 010405 (2010)
  • [52] R. Buniy, F. Colombo, I. Sabadini, D.C. Struppa, Quantum Harmonic Oscillator with superoscillating initial datum, J. Math. Phys., 55 (2014), 113511.
  • [53] T. Cheon, S. Poghosyan, Weak value expansion of quantum operators and its application in stochastic matrices, arXiv:1306.4767, 2013.
  • [54] A. Cho, Particle Physicists New Extreme Teams, Science, 333 (2011).
  • [55] Y.-W. Cho, H.-T. Lim, Y.-S. Ra, Y.-H. Kim, Weak Value Measurement with an Incoherent Measuring Device, New J. Phys. 12, 023036 (2010).
  • [56] F. Colombo, J. Gantner, D. C. Struppa, Evolution of superoscillations for Schrödinger equation in uniform magnetic field. Preprint 2016.
  • [57] F. Colombo, J. Gantner, D. C. Struppa, Evolution by Schrödinger equation of Ahronov-Berry superoscillations in centrifugal potential, Preprint 2017.
  • [58] F. Colombo, I. Sabadini, D. C. Struppa, An introduction to superoscillatory sequences, in Noncommutative analysis, operator theory and applications, Linear Operators and Linear Systems 252. New York, NY: Birkhäuser / Springer (2016), 97âĂŞ104.
  • [59] F. Colombo, D. C. Struppa, A. Yger, Superoscillating sequences towards approximation in or -type spaces and extrapolation, Preprint 2017.
  • [60] T. Denkmayr, H. Geppert, S. Sponar, H. Lemmel, A. Matzkin, J. Tollaksen, Y. Hasegawa, Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment, Nature Communications, 5, 2014.
  • [61] M. R. Dennis, Canonical representation of spherical functions: Sylvester’s theorem, Maxwell’s multipoles and Majorana’s sphere, J. Phys. A 37 (2004), no. 40, 9487–9500. MR 2095432,
  • [62] M. R. Dennis, A. Hamilton, J. Courtial, Optics Letters, 33 (2008), 2976-78.
  • [63] P. B. Dixon, D. J. Starling, A. N. Jordan, J. C. Howell, Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification, Phys. Rev. Lett. 102 (2009), 173601.
  • [64] P. B. Dixon, D. J. Starling, A. N. Jordan, J. C. Howell, Optimizing the signal-to noise ratio of a beam-deflection measurement with interferometric weak values, Phys. Rev. A 80, 041803(R) (2009).
  • [65] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
  • [66] Leon Ehrenpreis, Solution of some problems of division. I. Division by a polynomial of derivation, Amer. J. Math. 76 (1954), 883–903. MR 0068123,
  • [67] Leon Ehrenpreis, Solution of some problems of division. II. Division by a punctual distribution, Amer. J. Math. 77 (1955), 286–292. MR 0070048,
  • [68] Leon Ehrenpreis, Solutions of some problems of division. III. Division in the spaces, ${\scr D}’,{\scr H}, {\scr Q}_A,{\scr O}$, Amer. J. Math. 78 (1956), 685–715. MR 0083690,
  • [69] Leon Ehrenpreis, Fourier analysis in several complex variables, Pure and Applied Mathematics, Vol. XVII, Wiley-Interscience Publishers A Division of John Wiley & Sons, New York-London-Sydney, 1970. MR 0285849
  • [70] P. J. S. G. Ferreira, Superoscillations, in âĂIJNew Perspectives on Approximation and Sampling TheoryâĂİ. A. I. Zayed and G. Schmeisser eds., 247âĂŞ268.
  • [71] P. J. S. G. Ferreira, A. Kempf, Superoscillations: faster than the Nyquist rate, IEEE trans. Signal. Processing, 54 (2006), 3732-40.
  • [72] R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367–387. MR 0026940
  • [73] R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
  • [74] D. Gabor, Theory of communication, J. IEE, 93 (1946), 429âĂŞ457.
  • [75] Jean Gallier, Discrete mathematics, Universitext, Springer, New York, 2011. MR 2777371
  • [76] I. M. Gelfand, A. M. Yaglom, Integration in Functional Spaces, J. Math. Phys. 1, 48 (1960)
  • [77] R. E. George, L. M. Robledo, O. J. E. Maroney, M. S. Blokb, H. Bernien, M. L. Markham, D. J. Twitchen, J. J. L. Morton, A. D. Briggs, R. Hanson, Opening up three quantum boxes causes classically undetectable wavefunction collapse, PNAS 110, 3777 (2013)
  • [78] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O'Bryen, A. G. White, G. J. Pryde, Violation of the Leggett-Grag Inequality with Weak Measurements of Photons, Proc. Nat. Acad. Sci. 108, 1256 (2011).
  • [79] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
  • [80] Lucien Hardy, Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories, Phys. Rev. Lett. 68 (1992), no. 20, 2981–2984. MR 1163133,
  • [81] J. M. Hogan, J. Hammer, S.-W. Chiow, S. Dickerson, D. M. S. Johnson, T. Kovachy, A. Sugarbaker, M. A. Kasevich, Precision angle sensor using an optical lever inside a Sagnac interferometer, 2011, Opt. Lett., volume 36, pages 1698-1700.
  • [82] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer Verlag, Berlin Heidelberg, 1983.
  • [83] Achim Kempf and Paulo J. S. G. Ferreira, Unusual properties of superoscillating particles, J. Phys. A 37 (2004), no. 50, 12067–12076. MR 2106626,
  • [84] O. Hosten and P. Kwiat, Observation of the spin Hall effect of light via weak measurements, Science 319, 787 (2008).
  • [85] J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Interferometric Weak Value Deflections: Quantum and Classical Treatments, Phys. Rev. A 81, 033813 (2010).
  • [86] Kunio Ichinobe, Integral representation for Borel sum of divergent solution to a certain non-Kowalevski type equation, Publ. Res. Inst. Math. Sci. 39 (2003), no. 4, 657–693. MR 2025459
  • [87] A. Kaneko, Introduction to hyperfunctions, Mathematics and its Applications (Japanese Series), vol. 3, Kluwer Academic Publishers Group, Dordrecht; SCIPRESS, Tokyo, 1988. Translated from the Japanese by Y. Yamamoto. MR 1026013
  • [88] Goro Kato and Daniele C. Struppa, Fundamentals of algebraic microlocal analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 217, Marcel Dekker, Inc., New York, 1999. MR 1703357
  • [89] J. Kempe, Quantum Random Walks: an introductory overview, Contemporary Physics, 44 (2003), 307âĂŞ327; quant-ph/0303081.
  • [90] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Observation of Topologically Protected Bound States in Photonic Quantum Walks, Nat. Comm. 3, 882 (2012).
  • [91] T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, Exploring Topological Phases with Quantum Walks, Phys. Rev. A 82, 033429 (2010).
  • [92] J. M. Knight and L. Vaidman, Weak Measurement of Photon Polarization, Phys. Lett. A 143, 357 (1990).
  • [93] P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, T. Jennewein, Time-resolved double-slit interference pattern measurement with entangled photons, Phys. Rev. A 86, 012321 (2012).
  • [94] S. Kowalevski, Zur Theorie der partiellen Differentialeichungen, J. Reine Angew. Math., 80 (1875), 1âĂŞ32.
  • [95] M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, M. J. Padgett, Measurement of the light orbital angular momentum spectrum using an optical geometric transformation, Journal of Optics 13 (2011), 064006.
  • [96] Dae Gwan Lee and Paulo Jorge S. G. Ferreira, Superoscillations of prescribed amplitude and derivative, IEEE Trans. Signal Process. 62 (2014), no. 13, 3371–3378. MR 3245551,
  • [97] D. G. Lee, P. J. S. G. Ferreira, Superoscillations with optimal numerical stability, IEEE Sign. Proc. Letters 21 (12) (2014), 1443âĂŞ1447.
  • [98] K. Lee, D. C. Struppa, Some combinatorial identities arising from superoscillatory sequences, unpublished.
  • [99] B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
  • [100] C.-F. Li and X.-Y. Xu and J.-S. Tang and J.-S. Xu and G.-C. Guo, Ultrasensitive phase estimation with white light, 2011, Phys. Rev. A, volume 83, pages 044102.
  • [101] J. Lindberg, Mathematical concepts of optical superresolution, Journal of Optics 14 (2012) 083001.
  • [102] J. S. Lundeen, A. M. Steinberg, Experimental Joint Weak Measurement on a Phtoton Pair as a Probe of Hardy's Paradox, Phys. Rev. Lett. 102, 020404 (2009).
  • [103] D. A. Lutz, M. Miyake, and R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J. 154 (1999), 1–29. MR 1689170
  • [104] Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 271–355 (French). MR 0086990
  • [105] Stéphane Mallat, A wavelet tour of signal processing, Academic Press, Inc., San Diego, CA, 1998. MR 1614527
  • [106] R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, H. M. Wiseman, A Double-Slit âĂIJWhich-WayâĂİ Experiment on the Complementarity - Uncertainty Debate, New J. Phys. 9, 287 (2007).
  • [107] Elliott W. Montroll, Markoff chains, Wiener integrals, and quantum theory, Comm. Pure Appl. Math. 5 (1952), 415–453. MR 0052721,
  • [108] S. Nussinov, J. Tollaksen, Color Transparency in QCD and post-selection in quantum mechanics, Phys Rev D, 78 (2008), 036007.
  • [109] O. Oreshkov, T. A. Brun, Weak Measurements are Universal, Phys. Rev. Lett. 95 (2005), 110409.
  • [110] V. P. Palamodov, Linear differential operators with constant coefficients, Translated from the Russian by A. A. Brown. Die Grundlehren der mathematischen Wissenschaften, Band 168, Springer-Verlag, New York-Berlin, 1970. MR 0264197
  • [111] G. D. Paparo, V. Dunjko, A. Makmal, M. A. Martin-Delgado, H. J. Briegel, Quantum Speedup for Active Learning Agents, Phys. Rev. X 4, 031002 (2014).
  • [112] A. D. Parks, D. W. Cullin, D. C. Stoudt, Observation and Measurement of an Optical Aharonov-Albert-Vaidman Effect, Proc. R. Soc. Lond. 454, 2997 (1998).
  • [113] G. J. Pryde, J. L. O'Brien, A. G. White, T. C. Ralph, H. M. Wiseman, Measurement of quantum weak values of photon polarization, Phys. Rev. Lett., 94 (22) Art. No. 220405 JUN 10 2005.
  • [114] Wang Qiao, A simple model of Aharonov-Berry’s superoscillations, J. Phys. A 29 (1996), no. 9, 2257–2258. MR 1396434,
  • [115] A. A. G Requicha, The zeros of entire functions: Theory and engineering applications, Proc. IEEE 68 (3) (1980), 308âĂŞ328.
  • [116] K. J. Resch, J. S. Lundeen, A. M. Steinberg, Experimental Realization of the Quantum Box Problem, Physics Letters A, 324 vol.2-3, 125 (2004).
  • [117] N. W. M. Ritchie, J. G. Story, R. G. Hulet, Realization of a measurement of a âĂIJweak valueâĂİ, Phys. Rev.Lett. 66, 1107 (1991).
  • [118] E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, N. I. Zheludev, Nature Materials, 11 (2012), 432âĂŞ435.
  • [119] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • [120] J. J. Sakurai Modern Quantum Mechanics, Addison-Wesley Publishing Company, 1995.
  • [121] L. S. Schulman, Techniques and Applications of Path Integration, Dover, 2005.
  • [122] D. R. Solli, C. F. McCormik, R. Y. Chiao, S. Popescu, and J. M. Hickmann, Fast Light, Slow Light, and Phase Singularities: A Connection to Generalized Weak Values, Phys. Rev. Lett. 92, 043601 (2004).
  • [123] D. J. Starling, P. B. Dixon, A. N. Jordan, J. C. Howell, Optimizing the Signal-to-Noise Ratio of a Beam-Deflection Measurement with Interferometric Weak Values, Phys. Rev. A 80, 041803(R) (2009).
  • [124] D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Precision Frequency Measurements with Interferometric Weak Values, Phys. Rev. A 82, 011802(R) (2010).
  • [125] Daniele Carlo Struppa, The fundamental principle for systems of convolution equations, Mem. Amer. Math. Soc. 41 (1983), no. 273, iv+167. MR 683420,
  • [126] D. Suter, Weak Measurements and the Quantum Time-Translation Machine in a Classical System, Phys. Rev. A 51, 45 (1995).
  • [127] D. Suter, M. Ernst, R. R. Ernst, Quantum Time-Translation Machine - An Experimental Realization, Mol. Phys. 78, 95 (1993).
  • [128] B. A. Taylor, Some locally convex spaces of entire functions, Proc. Symp. Pure Math., 11 (1968), 431âĂŞ467.
  • [129] J. Tollaksen, Quantum Reality and Nonlocal Aspects of Time, 2001 PhD thesis, Boston University, ISBN 978 054 941 2946.
  • [130] J. Tollaksen, Non-statistical weak measurements, in Quantum Information and Computation V, E. Donkor, A. Pirich, H. Brandt (eds), Proc. of SPIE Vol. 6573 (SPIE, Bellingham, WA, 2007), CID 6573-33.
  • [131] J. Tollaksen, Robust Weak Measurements on Finite Samples, J. Phys. Conf. Series, vol. 70, (2007), 012015, quant-ph/0703038.
  • [132] J. Tollaksen, Y. Aharonov, A. Casher, T. Kaufherr, S. Nussinov,Quantum interference experiments, modular variables and weak measurements, New Journal of Physics, 12 (2010), 013023.
  • [133] M. D. Turner, C. A. Hagedorn, S. Schlamminger, J. H. Gundlach, Picoradian deflection measurement with an interferometric quasi-autocollimator using weak value amplification, 2011, Opt. Lett., volume 36, 1479âĂŞ1481.
  • [134] V. Vladimirov, Distributions en physique mathématique, “Mir”, Moscow, 1979 (French). Translated from the Russian by Irina Petrova. MR 555245
  • [135] John von Neumann, Mathematical foundations of quantum mechanics, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1996. Translated from the German and with a preface by Robert T. Beyer; Twelfth printing; Princeton Paperbacks. MR 1435976
  • [136] Q. Wang, F.-W. Sun, Y.-S. Zhang, Jian-Li, Y.-F. Huang, G. C. Guo, Experimental Demonstration of a Method to Realize Weak Measurement of the Arrival Time of a Single Photon, Phys. Rev. A 73, 023814 (2006).
  • [137] H. M. Wiseman,Weak values, quantum trajectories, and the cavity-QED experiment on wave-particle correlation, Phys. Rev. A, 65 (2002), Art. No. 032111.
  • [138] A. M. Yao, M. J. Padgett, Orbital angular momentum: origins, behavior and applications, Advances in Optics and Photonics 3, (2011), 161âĂŞ204.
  • [139] K. Yokota, T. Yamamoto, M. Koashi, N. Imoto, Direct Observation of Hardy's paradox by joint weak measurement with an entangled photon pair, New J. Phys. 11, 033011 (2009).
  • [140] Kôsaku Yosida, Functional analysis, 5th ed., Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Band 123. MR 0500055
  • [141] O. Zilberberg, A. Romito, Y. Gefen, Charge sensing amplification via weak values measurement, Phys. Rev. Lett. 106, 080405 (2011) arXiv:1009.4738.