The discrete fractional Fourier transform and Harper's equation
Open accessAbstract
It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
For N odd, the vector e_n with N entries defined from the eigenvector E_n with n zero crossings will, with proper normalization, contain approximations of the sample values of the Gauss-Hermite function ψ_n, and the vectors e_n converge to the vectors Ψ_n as N→∞.
“For N odd, define the vector with N entries”
From Computation of the fractional Fourier transform (2004) · cites this paper as [3]
Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredFor N even, the vector e_n with N+1 entries defined from the eigenvector E_n with n zero crossings will, with proper normalization, contain approximations of the sample values of the Gauss-Hermite function ψ_n, and the vectors e_n converge to the vectors Ψ_n as N→∞.
“For N even, define the vector with N +1 entries as”
From Computation of the fractional Fourier transform (2004) · cites this paper as [3]
Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
The algebraic eigenvalue problem
Wilkinson, J. H. (1965) · Cited by 5,833 · see citing works
The Algebraic Eigenvalue Problem
I., E., & Wilkinson, J. H. (1966) · Mathematics of Computation · doi:10.2307/2003558 · Cited by 5,208 · see citing works
Proceedings IEEE International Conference on Acoustics Speech and Signal Processing
Zheng, F., McLaughlin, S., & Mulgrew, B. (1991) · Cited by 1,906 · see citing works
The Fractional Order Fourier Transform and its Application to Quantum Mechanics
NAMIAS, V. (1980) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/25.3.241 · Cited by 1,722 · see citing works
The Fractional Fourier Transform: with Applications in Optics and Signal Processing
Özaktaş, H. M., Kutay, M. A., & Zalevsky, Z. (2001) · International Linear Collider · Cited by 1,385 · see citing works
On Namias's Fractional Fourier Transforms
McBRIDE, A. C., & KERR, F. H. (1987) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/39.2.159 · Cited by 597 · see citing works
Eigenvalue and eigenvector decomposition of the discrete Fourier transform
McClellan, J., & Parks, T. (1972) · IEEE Transactions on Audio and Electroacoustics · doi:10.1109/tau.1972.1162342 · Cited by 257 · see citing works
Improved discrete fractional Fourier transform
Pei, S.-C., & Yeh, M.-H. (1997) · Optics Letters · doi:10.1364/ol.22.001047 · Cited by 188 · see citing works
Advances in Imaging and Electron Physics
Hawkes, P. (1998) · Cited by 129 · see citing works
An algebraic semi-classical approach to Bloch electrons in a magnetic field
Rammal, R., & Bellissard, J. (1990) · Journal de physique · doi:10.1051/jphys:0199000510170180300 · Cited by 90 · see citing works
The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform
Barker, L., Candan, Ç., Hakioglu, T., Kutay, M. A., & Ozaktas, H. M. (2000) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/33/11/304 · Cited by 76 · see citing works
Self-citationDiscrete-time, discrete-frequency, time-frequency analysis
Richman, M. S., Parks, T. W., & Shenoy, R. G. (1998) · IEEE Transactions on Signal Processing · doi:10.1109/78.678465 · Cited by 60 · see citing works
Holomorphic quantization on the torus and finite quantum mechanics
Athanasiu, G. G., Floratos, E. G., & Nicolis, S. (1996) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/29/21/010 · Cited by 38 · see citing works
Coherent states in Finite Quantum Mechanics
Athanasiu, G. G., & Floratos, E. G. (1994) · Nuclear Physics B · doi:10.1016/0550-3213(94)90184-8 · Cited by 35 · see citing works
Hilbert Space Approach to Some Classical Transforms
Picard, R. H., & Picard, R. H. (1989) · Medical Entomology and Zoology · Cited by 30 · see citing works
Holomorphic quantization on the torus and finite quantum mechanics
Athanasiu, G. G., Floratos, E. G., & Nicolis, S. (1995) · CERN Document Server (European Organization for Nuclear Research) · Cited by 27 · see citing works
Continuum Quantum Systems as Limits of Discrete Quantum Systems, I: State Vectors
Barker, L. (2001) · Journal of Functional Analysis · doi:10.1006/jfan.2001.3788 · Cited by 14 · see citing works
Self-citationLinear canonical transformations and quantum phase: a unified canonical and algebraic approach
Hakioglu, T. (1999) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/32/22/312 · Cited by 9 · see citing works
Linear canonical transformations and quantum phase: a unified canonical and algebraic approach
Hakioglu, T. (1999) · CERN Document Server (European Organization for Nuclear Research) · Cited by 8 · see citing works
Flagged first, then by citations
Research works printed but not linked: 5
- N. M. Atakishiev and S. K. Suslov. Difference analogues of the harmonic oscillator. (English translation.) Teoreticheskaya i Mat. Fiz., 85 (1990), 64-73.
- R. Balian and C. Itzykson. Observations sur la mecanique quantique finie. C. R. Acad. Sc. Paris, Ser. I. 303 (1986), 773-778.
- L. Barker. Phase space over an abelian group of odd order (preprint).
- Q. Candan. The Discrete Fractional Fourier Transform (Bilkent University M.S. thesis, July 1998, unpublished).
- A. C. McBride and F. H. Kerr. On Namias' fractional Fourier transform. IMA J. of Applied Math., 39 (1987), 159 175.
Cited by
Sorted by citations
The discrete fractional Fourier transform
Candan, C., Kutay, M. A., & Ozaktas, H. M. (2000) · IEEE Transactions on Signal Processing · doi:10.1109/78.839980 · Cited by 683 · see citing works
Computation of the fractional Fourier transform
Bultheel, A., & Martı́nez Sulbaran, H. E. (2004) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2004.02.001 · Cited by 150 · see citing works
Recent developments in the theory of the fractional Fourier and linear canonical transforms
Bultheel, A., & Sulbaran, H. M. (2007) · Bulletin of the Belgian Mathematical Society - Simon Stevin · doi:10.36045/bbms/1170347822 · Cited by 83 · see citing works
The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform
Barker, L., Candan, Ç., Hakioglu, T., Kutay, M. A., & Ozaktas, H. M. (2000) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/33/11/304 · Cited by 76 · see citing works
Self-citationA Survey of Signal Processing Problems and Tools in Holographic Three-Dimensional Television
Onural, L., Gotchev, A. P., Özaktaş, H. M., & Stoykova, E. (2007) · IEEE Transactions on Circuits and Systems for Video Technology · doi:10.1109/tcsvt.2007.909973 · Cited by 64 · see citing works
The discrete fractional Fourier transform
Candan, C., Kutay, M. A., & Ozaktas, H. M. (1999) · 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258) · doi:10.1109/icassp.1999.756324 · Cited by 52 · see citing works
General phase spaces: from discrete variables to rotor and continuum limits
Albert, V. V., Pascazio, S., & Devoret, M. (2017) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aa9314 · Cited by 34 · see citing works
Hanna, M. T., Seif, N. P. A., & Ahmed, W. (2008) · Digital Signal Processing · doi:10.1016/j.dsp.2008.05.003 · Cited by 34 · see citing works
A shattered survey of the Fractional Fourier Transform
Bultheel, A. (2002) · Lirias · Cited by 27 · see citing works
Continuum quantum systems as limits of discrete quantum systems: II. State functions
Barker, L. (2001) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/34/22/308 · Cited by 15 · see citing works
Self-citationContinuum Quantum Systems as Limits of Discrete Quantum Systems, I: State Vectors
Barker, L. (2001) · Journal of Functional Analysis · doi:10.1006/jfan.2001.3788 · Cited by 14 · see citing works
Self-citationContinuum quantum systems as limits of discrete quantum systems. III. Operators
Barker, L. (2001) · Journal of Mathematical Physics · doi:10.1063/1.1398582 · Cited by 11 · see citing works
Self-citationThe Algorithm of Fractional Fourier Transform and Application in Digital Image Encryption
Zhang, Y., & Zhao, F. (2009) · International Conference on Information Engineering and Computer Science · doi:10.1109/iciecs.2009.5365644 · Cited by 9 · see citing works
On the raising and lowering difference operators for eigenvectors of the finite Fourier transform
Atakishiyeva, M. K., & Atakishiyev, N. M. (2015) · Journal of Physics Conference Series · doi:10.1088/1742-6596/597/1/012012 · Cited by 6 · see citing works
The Algorithm and Error Analysis of Fractional Fourier Transform
Zhao, F., & Zhai, X. (2010) · International Conference on Information Engineering and Computer Science · doi:10.1109/iciecs.2010.5678385 · Cited by 2 · see citing works
Implementación Computacional de la Transformada Fraccional de Fourier Discreta
Rojas, H. E., Cortes, C. A., & Ramírez, D. F. (2014) · Información tecnológica · doi:10.4067/s0718-07642014000600017 · Cited by 1 · see citing works
Recent developments in the theory of fractional transforms
Bultheel, A., & Sulbaran, H. M. (2003) · Lirias (KU Leuven)
- Authors
- Barker, L.
- Affiliations
- Bilkent University (TR)
- Corresponding author
- Barker, Laurence
- ORCID
- Barker, Laurence
- Venue
- Mathematika
- Publisher
- Wiley
- Published
- 2000-12
- Volume
- 47
- Issue
- 1-2
- Pages
- 281-297
- ISSN
- 0025-5793 (ISSN-L)
- 2041-7942
- OpenAlex
- W1985388437
- Cited by
- 17
- References
- 19
- Citations by year
- 2017: 1 · 2015: 1 · 2014: 1
- FWCI
- 5.01
- Type
- journal article
- Language
- en
- Open access
- green
- hdl.handle.net
- Licence
- Locations
- Mathematika (publishedVersion)
- Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Physical Sciences › Mathematics › Modeling and Simulation › Fractional Differential Equations Solutions
- Keywords
- discrete fractional Fourier transform, limit processes, inner product
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
