The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform
Open accessAbstract
Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.
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TL;DR
The paper defines Harper functions as solutions to Harper’s equation that form an orthonormal basis of discrete analogues of Hermite-Gaussian functions, and uses them to define a discrete fractional Fourier transform. It shows that this transform and the associated standard discrete harmonic oscillator converge to their continuum counterparts in time evolution, with numerical evidence for small cases.
Key findings
- The Harper functions \(h_{n,k}\) are the unique basis, up to real factors, simultaneously diagonalizing \(F_n\) and \(G_n\); they solve Harper’s equation \(h(X-1)+2\cos(2\pi X/n)h(X)+h(X+1)=\lambda h(X)\), and converge to Hermite-Gaussians \(h_k\) for suitable sequences of \(n\) (Proposition 3).
- The discrete fractional FT \(F_n^{[t]}\) is defined by \(F_n^{[t]}h_{n,k}=e^{2\pi ikt}h_{n,k}\); for suitable sequences it converges to the continuum fractional FT \(F_\infty^{[t]}\), and \(F_n\) converges to the continuum FT \(F_\infty\) (Proposition 4). A variant \(F_n^{(t)}=e^{2\pi iK_n t}\) has the same eigenvectors and eigenvalues differing by \(o(1/n)\), and also converges to \(F_\infty^{[t]}\) (Propositions 5–6).
- The standard discrete harmonic oscillator has Hamiltonian \(H_n=\frac{n}{2\pi}(4-G_n)=2K_n+1\), with time evolution \(S_n(t)=e^{-iH_nt}=e^{-it}F_n^{(-t/\pi)}\). Although the Hamiltonians \(H_n\) do not converge to the continuum \(H_\infty\), the time evolutions \(S_n(t)\) converge to \(S_\infty(t)\) for suitable sequences (Proposition 9).
- Numerical comparisons show convergence “looks fairly good”: for \(n=16\), the first six Harper functions are compared with Hermite-Gaussians; for \(10\le n\le 40\) and \(0\le k\le 5\), \(L^2\)-norm differences are plotted; for \(25\le n\le 100\) and \(0\le k\le 9\), \(\eta(n,k)\) appears to converge to \(k\).
Methods
- Finite-dimensional algebraic setting: \(L(n)\) is the space of functions \(\mathbb Z/n\to\mathbb C\); Harper’s equation is treated as the eigenproblem for the Hermitian operator \(G_n\). A tridiagonal matrix crossing-number result (Proposition 1) is used to define and order the Harper functions.
- The discrete fractional FT is constructed by assigning eigenvalues to the common eigenvectors of \(F_n\) and \(G_n\), following Pei and Yeh; convergence to continuum operators is defined via sequences of vectors/operators from prior work.
- Numerical section compares Harper functions with Hermite-Gaussians for \(n=16\), plots \(L^2\) differences for \(10\le n\le 40\), \(0\le k\le 5\), and plots \(\eta(n,k)\) for \(25\le n\le 100\), \(0\le k\le 9\).
Limitations
- Stated: no closed formula for the Harper functions is known, described as a disadvantage.
- Stated: the arguments provide no information on the accuracy of Harper functions as approximations to Hermite-Gaussians, so section 5 gives numerical information only.
- Stated: it is unresolved whether \(F_n^{[1/4]}=F_n\); the paper conjectures an affirmative answer. Also, convergence results are for “suitable infinite sequences” of \(n\), not all \(n\). Evident: numerical evidence is limited to small \(n\) and \(k\) (e.g., \(n=16\), \(n\le 40\), \(k\le 5\), and \(n\le 100\), \(k\le 9\)) with no error bounds.
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Claims the paper makes
0 of 10 cited sources checked · 0 supported · 0 flagged
- Findingnone given
For an integer k ≥ 0, writing μ(n,k) for the (k+1)-th largest eigenvalue of G_n, for suitable infinite sequences of positive integers n, μ(n,k) n = 4 - 2π(2j+1)/n + o(1/n).
“Proposition 5. Given an integer k 0, and writing µ(n, k) for the k + 1th largest eigenvalue of G n , then for suitable infinite sequences of positive integers n, we have µ(n, k) n = 4 -2π(2j + 1)/n + o(1/n).”
Quote in the paperNot a substantial claim - Findingreasoning
If two Harper functions have the same eigenvalue, then either they are the same index or their indices have opposite parity.
“If λ n,k = λ n,k , then either k = k or else k and k have opposite parity.”
Quote in the paperNot a substantial claim - Findingreasoning
Up to multiplication by real factors, the Harper functions form the unique basis of L(n) that simultaneously diagonalizes F^2 and G.
“Proposition 2. Up to multiplication by real factors, the vectors h n,k comprise the unique basis of L(n) simultaneously diagonalizing F 2 and G.”
Quote in the paperSubstantial claim - Findingreasoning
The convergence of time evolutions for all t is equivalent to the convergence of states for all t given convergent initial states.
“Proposition 7. Let us fix a sequence of positive integers n.Then the following two conditions are equivalent: (a) The time evolutions S n (t) converge to the time evolution S ∞ (t) for all t.(b) Given any initial states φ n (0) converging to an initial state φ ∞ (0), then φ n (t) converges to φ ∞ (t) for all t.”
Quote in the paperSubstantial claim - Findingcitation
For a real symmetric tridiagonal matrix A with positive off-diagonal entries, the eigenvector corresponding to the k-th largest eigenvalue has crossing number k.
“Proposition 1.For the real symmetric tridiagonal matrix A as above, and an integer k with 0 k r, the eigenvector v k has crossing number k.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [33] source
Not checked in the free run.
- Findingreasoning
Replacing X with W and applying a Galois automorphism recovers Harper's equation from the Bloch electron problem.
“Replacing X with a variable W such that mX + k = mW , then replacing φ with its image θ under a Galois automorphism such that e 2π im/n → e 2π i/n , we recover Harper's equation θ(W -1) + θ(W + 1) + 2 cos(2π W/n)θ (W ) = Eθ (W ).”
Quote in the paperNot a substantial claim - Findingreasoning
The Harper functions form an orthonormal basis of solutions to Harper's equation.
“The Harper functions form an orthonormal basis of solutions to Harper's equation: h n,k (X -1) + 2 cos(2πX/n)h n,k (X) + h n,k (X + 1) = λ n,k h n,k (X).”
Quote in the paperSubstantial claim - Findingreasoning
The Harper functions are precisely those vectors with signs and an ordering suitably chosen.
“These vectors, with signs and an ordering suitably chosen, are precisely the Harper functions.”
Quote in the paperNot a substantial claim - Findingreasoning
The time evolution of the standard discrete harmonic oscillator is S_n(t) = e^{-iH_n t} = e^{-i(2K_n+1)t} = e^{-it} F^{(-t/π)}_n.
“The time evolution of this quantum system is S n (t) = e -iH n t = e -i(2K n +1)t = e -it F (-t/π) n .”
Quote in the paperSubstantial claim - Findingreasoning
The time evolution of the standard continuum harmonic oscillator is S_∞(t) = e^{iH_∞ t} = e^{-it} F^{[-t/π]}_∞.
“Therefore, the time evolution for this quantum system is S ∞ (t) = e iH ∞ t = e -it F [-t/π] ∞ .”
Quote in the paperSubstantial claim- Rests on an assumption: The standard continuum harmonic oscillator's energy eigenfunctions satisfy H_∞ h_k = (2k+1)h_k.
- Findingown data
For n = 16, the first six Harper functions compared with the first six Hermite-Gaussians show that for k ≤ 3 the convergence already looks fairly good.
“In figure 1, with n = 16, the first six Harper functions h 16,0 , . . ., h 16,5 (indicated by circles) are compared with the first six Hermite-Gaussians (indicated by the curves).The period n = 16 is usually too small for useful calculation; the point is that, for this small value of n, and for k 3, the convergence already looks fairly good.”
Quote in the paperNot a substantial claim - Findingnone given
The construction of the Harper functions reduces Harper's equation to two independent systems of equations, each being the eigenvector problem for a real symmetric tridiagonal matrix.
“We mention that our construction of the Harper functions, in effect, reduces Harper's equation to two independent systems of equations, each of which is the eigenvector problem for a real symmetric tridiagonal matrix.”
Quote in the paperSubstantial claim - Findingreasoning
Due to ill-conditioning, the recursive method for calculating eigenvectors is not an effective method for calculating the Harper functions numerically.
“However, due to ill-conditioning, this is not an effective method for calculating the Harper functions numerically.”
Quote in the paperNot a substantial claim - Premisecitation
For any integer k ≥ 0, for suitable infinite sequences of positive integers n, the sequence of Harper functions h_{n,k} converges to the k-th Hermite-Gaussian function h_k.
“Proposition 3. Consider an integer k 0. For suitable infinite sequences of positive integers n, the sequence (h n,k ) n converges to h k .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [6] Ballesteros A, 1998
Not checked in the free run.
- Interpretationnone given
The discrete fractional Fourier transform is essentially the time-evolution operator of the discrete harmonic oscillator.
“The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.”
Quote in the paperSubstantial claim - Premisecitation
For suitable infinite sequences of positive integers n, the discrete fractional Fourier transform F^[t]_n converges to the continuum fractional Fourier transform F^[t]_∞.
“Proposition 4. For suitable infinite sequences of positive integers n, the discrete fractional FT F [t] n converges to the continuum fractional FT F [t] ∞ .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [6] Ballesteros A, 1998
Not checked in the free run.
- Interpretationreasoning
For suitable infinite sequences of positive integers n, the time evolution S_n(t) corresponding to H_n converges to the time evolution S_∞(t) corresponding to H_∞.
“Proposition 9.The Hamiltonians H n and H of the standard harmonic oscillators are such that, for suitable infinite sequences of positive integers n, the time evolution S n (t) corresponding to H n converges to the time evolution S ∞ (t) corresponding to H ∞ .”
Quote in the paperSubstantial claim- Rests on an assumption: The standard continuum harmonic oscillator's energy eigenfunctions satisfy H_∞ h_k = (2k+1)h_k.
- Interpretationnone given
The algebraic felicity of the discrete fractional Fourier transform makes it especially amenable to interpretation as time-evolution operators of harmonic oscillators.
“the algebraic felicity of the discrete fractional FT, as we define it, makes this discrete fractional FT especially amenable to such an interpretation.”
Quote in the paperNot a substantial claim - Interpretationnone given
Harper functions are natural algebraic analogues of the Hermite-Gaussian functions.
“in an algebraic sense that will be made clear in section 2, they are natural analogues of the Hermite-Gaussian functions.”
Quote in the paperSubstantial claim - Premisecitation
For any infinite sequence of positive integers n and any Hermitian operator H_∞ on L^2(R), there exist Hermitian operators H_n on L(n) such that the equivalent conditions of Proposition 7 hold.
“Proposition 8. Consider any infinite sequence of positive integers n.Let H ∞ be any Hermitian operator (with a dense domain) on L 2 (R).Then there exist Hermitian operators H n on L(n) such that the equivalent conditions (a) and (b) in proposition 7 hold.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [7] Barker L J, 2000
Not checked in the free run.
- Interpretationreasoning
For suitable infinite sequences of positive integers n, the operator F^(t)_n converges to the continuum fractional Fourier transform F^[t]_∞.
“Proposition 6. For suitable infinite sequences of positive integers n, the operator F (t) n converges to the continuum fractional FT F [t] ∞ .”
Quote in the paperNot a substantial claim - Interpretationreasoning
The discrete fractional Fourier transform provides a good numerical approximation to the time evolution of the continuum harmonic oscillator.
“This may be interpreted as saying that the discrete fractional FT provides a good numerical approximation to the time evolution of the continuum harmonic oscillator.”
Quote in the paperSubstantial claim - Interpretationreasoning
The discrete fractional Fourier transform as an approximation is good for functions whose coefficients c_k converge quickly to zero as k increases.
“Thus the discrete fractional FT, as an approximation, is good for those functions whose coefficients c k converge quickly to zero as k increases.”
Quote in the paperSubstantial claim - Interpretationown data
The discrete fractional Fourier transform is a good numerical approximation to the continuum fractional Fourier transform.
“In conclusion, we have given evidence that the discrete fractional FT is a good numerical approximation to the continuum fractional FT.”
Quote in the paperSubstantial claim - Interpretationreasoning
For a fixed square-integrable function f, if n is large enough that for each k at least one of |c_k| or the norm difference is negligible, then F^[t]_n f_n and F^(t)_n are approximately equal to the sample values of F^[t]_∞ f.
“For fixed f , if n is chosen large enough to ensure that, for each k, at least one of |c k | or h n,kh k 2 is negligible, then F [t] n f n and F (t) n will be approximately equal to the vector of sample values of F [t] ∞ f .”
Quote in the paperSubstantial claim - Interpretationreasoning
For large n, F^[1/4]_n is approximately equal to F_n.
“Proposition 4 tells us that F [1/4] n is approximately equal to F n for large n.”
Quote in the paperNot a substantial claim - Premisecitation
Assuming ω is rational, the energy eigenstates of the Bloch electron problem are given by ψ(X,Y) = e^{2πikY/n} φ(X), where φ solves a difference equation.
“Assuming that ω is rational, let us write write ω = m/n where m is an integer, and n is a positive integer coprime to m.The energy eigenstates ψ with energy eigenvalue E are given by ψ(X, Y ) = e 2π ikY/n φ(X) where k is an integer, and φ is a solution to φ(X -1) + φ(X + 1) + 2 cos(2π(mX + k)/n)φ(X) = Eφ(X).”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation
There exists, uniquely up to a choice of sign, an orthonormal basis of real unit vectors simultaneously diagonalizing F_n and G_n.
“However, as observed in [12], there does exist, uniquely up to a choice of sign, an orthonormal basis of real unit vectors simultaneously diagonalizing F n and G n .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] Cohen L, 1995
Not checked in the free run.
- Interpretationreasoning
The continuum constructions are realized as limits of the discrete constructions.
“Furthermore, the continuum constructions are realized as limits of the discrete constructions.”
Quote in the paperSubstantial claim
- Not yet classified. 5
- The quote only partly states this claim. 1
- May combine two claims. 4
- Classified as probably not a substantial claim. 5
Not yet classified.
FindingWith respect to the basis E+, the coordinates of h_{n,2j} have crossing number j; with respect to E-, the coordinates of h_{n,2j+1} have crossing number j.
“With respect to the basis E + , the coordinates of h n,2j have crossing number j .With respect to the basis E -, the coordinates of h n,2j +1 have crossing number j .”
Quote in the paperNot yet classifiedNot yet classified.
FindingThe Hamiltonians H_n of the standard discrete harmonic oscillator do not converge to the continuum harmonic oscillator Hamiltonian H_∞.
“The operators H n do not converge to H ∞ .”
Quote in the paperNot yet classifiedThe quote only partly states this claim.
Findingnone givenHarper functions are discrete approximations to the Hermite-Gaussian functions.
“Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataThe numerical data in section 5 indicates the accuracy of the discrete fractional Fourier transform as an approximation to the continuum fractional Fourier transform and the accuracy of the discrete harmonic oscillator as an approximation to the continuum harmonic oscillator.
“The numerical data in section 5 may be viewed in two ways: it indicates the accuracy of the discrete fractional FT as an approximation to the continuum fractional FT; it indicates the accuracy of the discrete harmonic oscillator as an approximation to the continuum harmonic oscillator.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone givenThe techniques used in the arguments provide no information on the accuracy of the Harper functions as approximations to the Hermite-Gaussians.
“The techniques used in the arguments above appear to provide no information on the accuracy of the Harper functions as approximations to the Hermite-Gaussians.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingown dataFigure 2 shows the L2 norm of the difference between Harper functions and Hermite-Gaussians for 10 ≤ n ≤ 40 and 0 ≤ k ≤ 5, and the convergence looks fairly good.
“Figure 2 shows, for 10 n 40 and 0 k 5, the L 2 -norm h n,kh k 2 of the difference between the Harper function h n,k and the Hermite-Gaussian h k .The difference was calculated by evaluating h n at the sample points, normalizing, and comparing with the vector h n,k .Again, the convergence looks fairly good.”
Quote in the paperNot a substantial claimNot yet classified.
FindingF^[t]_n and F^(t)_n are not the same, but their discrepancy diminishes quickly as n → ∞.
“Although F [t] n and F (t) n are not the same, we shall see in section 5 that the discrepancy diminishes quickly as n → ∞.”
Quote in the paperNot yet classifiedMay combine two claims.
FindingreasoningThe Harper functions h_{n,k} are eigenvectors of both F^(t)_n and F^[t]_n, and the corresponding eigenvalues are the same up to o(1/n).
“Thus, the Harper functions h n,k are the eigenvectors of F (t) n and F [t] n ; the corresponding eigenvalues are the same up to o(1/n).”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingown dataEmpirical confirmation that η(n,k) converges to k is given by the graphs in figure 3.
“Empirical confirmation that η(n, k) converges to k is given by the graphs, in figure 3, of (n, k) := 1η(n, k)/k against n, where 25 n 100 and 0 k 9.”
Quote in the paperNot a substantial claimNot yet classified.
PremiseThe discrete Fourier transform F_n converges to the continuum Fourier transform F_∞.
“Furthermore, the discrete FT F n converges to the continuum FT F ∞ .”
Quote in the paperNot yet classified0 of 1 sources checked- cites [6] Ballesteros A, 1998
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe discrete fractional Fourier transform discussed in the paper performs numerical calculation of the continuum Fourier transform.
“the discrete fractional FT we discuss does indeed perform this service; see [10].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [10] Candan Ç, 1999
Not checked in the free run.
Not yet classified.
InterpretationThe question of whether F^[1/4]_n equals F_n is equivalent to asking whether h_{n,k}, as an eigenvalue of F_n, always has eigenvalue i^k.
“The question is equivalent to asking whether or not h n,k , as an eigenvalue of F n , always has eigenvalue i k .”
Quote in the paperNot yet classifiedProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe eigenvalues of G_n are not always distinct.
“The eigenvalues of G n are not always distinct; Dickinson-Steiglitz [12] conjecture that the eigenvalues are distinct save that, when 4 divides n, the eigenvalue 0 has multiplicity 2.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [12] Cohen L, 1995
Not checked in the free run.
May combine two claims.
InterpretationreasoningThe discrete fractional Fourier transform is an algebraic analogue of the continuum fractional Fourier transform and provides an algebraic analogue of the continuum harmonic oscillator.
“At least as importantly, the discrete fractional FT is also an algebraic analogue of the continuum fractional FT, and provides an algebraic analogue of the continuum harmonic oscillator.”
Quote in the paperSubstantial claim
The paper relies on this without support.
The standard continuum harmonic oscillator's energy eigenfunctions satisfy H_∞ h_k = (2k+1)h_k.
“It is well known that the solutions to the Schrödinger equation of the standard continuum harmonic oscillator are H ∞ h k = (2k + 1)h k .”
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 [4]
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 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 [4]
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 measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Explicit Hamiltonian inserts an imaginary unit into the cosine
MinorThe expanded action of H n uses cos(2π iX/n), whereas the stated definition H n = n(4 − G n)/2π and Harper's equation use cos(2πX/n). These are different diagonal terms.
“H n [X] = n 2π (-[X -1] + (4 -2 cos(2π iX/n))[X] -[X + 1]).”
“h n,k (X -1) + 2 cos(2πX/n)h n,k (X) + h n,k (X + 1) = λ n,k h n,k (X).”
Wootters citation points to Wolf in the reference list
MinorThe cited reference [35] is listed as a work by Wolf; the Wootters paper is listed as reference 36.
“Wootters [35] suggested the term discrete quantum system”
“35. Wolf K B 1979 Integral Transforms in Science and Engineering (New York: Plenum)”
η(n, k) is called an eigenvector but used as a scalar eigenvalue
MinorThe text introduces η(n, k) as an eigenvector of K n, then equates it to a scalar approaching k and uses it in a scalar exponential.
“Let η(n, k) be the k + 1th smallest eigenvector of the linear map”
“By proposition 5, η(n, k) = n(2µ(n, k)/2)/2π -1/2 = k + o(1).”
Continuum time-evolution exponential has the opposite sign
MinorThe paper defines time evolution with a negative exponent, but later writes a positive exponent for the continuum oscillator. The latter also conflicts with the negative-exponent expression on the same line.
“S ∞ (t) = e iH ∞ t = e -it F [-t/π] ∞ .”
“The time evolution S of the system is defined to be the group homomorphism R → U(V ) given by S(t) := e -iHt .”
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Hakioglu, T. (1999) · CERN Document Server (European Organization for Nuclear Research) · Cited by 8 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 3
- Atakishiyev N M, 1990, Teor. Mat. Fiz., 85
- Candan Ç, 1999, 3, 1713
- Candan Ç, IEEE Trans. Signal Process.
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
Self-citationComputation of the fractional Fourier transform
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“In both cases one has to take care that if we compute F af for a ranging over the interval [0; 4 ],”
BackgroundThe fractional discrete cosine transform
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Recent developments in the theory of the fractional Fourier and linear canonical transforms
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Finite two-dimensional oscillator: I. The Cartesian model
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Sampling and series expansion theorems for fractional Fourier and other transforms
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Self-citationA Survey of Signal Processing Problems and Tools in Holographic Three-Dimensional Television
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Uncertainty Relation for the Discrete Fourier Transform
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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
Self-citationCONTRACTION OF THE FINITE ONE-DIMENSIONAL OSCILLATOR
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Optimal Fractional Fourier Filtering for Graph Signals
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General phase spaces: from discrete variables to rotor and continuum limits
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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
Geometry and dynamics in the fractional discrete Fourier transform
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Discrete Linear Canonical Transform Based on Hyperdifferential Operators
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Fractional Fourier analysis using the mobius inversion formula
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A Comprehensive Survey on Fractional Fourier Transform
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A shattered survey of the Fractional Fourier Transform
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Rotation and gyration of finite two-dimensional modes
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Digital Images Phase Encryption Using Fractional Fourier Transform
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Signal recovery from partial fractional Fourier domain information and its applications
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The discrete fractional Fourier transform and Harper's equation
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Development of Linear Canonical Transforms: A Historical Sketch
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New definition of the discrete fractional Fourier transformPreprint
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XFT: A Discrete Fourier Transform
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- Authors
- Barker, L., Candan, Ç., Hakioglu, T., Kutay, M. A., & Ozaktas, H. M.
- Affiliations
- Bilkent University (TR)
- Drexel University (US)
- Venue
- Journal of Physics A: Mathematical and General
- Publisher
- IOP Publishing
- Published
- 2000-03-10
- Volume
- 33
- Issue
- 11
- Pages
- 2209-2222
- ISSN
- 0305-4470 (ISSN-L)
- 1361-6447
- OpenAlex
- W2096438497
- Semantic Scholar
- 0761798759ec9c599eb177140b5b3d79dd9fdff0
- Cited by
- 76
- References
- 31
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 1 · 2023: 3 · 2021: 1 · 2020: 1 · 2019: 7 · 2018: 1 · 2017: 3 · 2016: 2 · 2015: 3 · 2014: 2 · 2013: 6 · 2012: 5
- FWCI
- 12.18
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Physics A Mathematical and General (publishedVersion)
- Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
- OpenMETU (Middle East Technical University) (submittedVersion, cc-by-nc-nd)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Keywords
- discrete fractional Fourier transform, Hermite-Gaussian functions, energy eigenfunctions, evolution operator, quasiperiodic lattices
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