Pith. sign in

REVIEW 3 major objections 4 minor 48 references

Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that random phaseless sampling at density 3 succeeds for complex-generated shift-invariant spaces when the generator satisfies a generalized Haar condition, and gives a constructive recursive reconstruction algorithm.

desk verdict The random-sampling theorems are plausible and worth a serious referee, but the paper ships with a demonstrably wrong proof in its motivating Theorem 1.1 and a key hypothesis (GHC) that has no rigorously verified nonempty domain in the complex case. read the letter →

arxiv 1908.05423 v5 pith:GQ6ET6HW submitted 2019-08-15 cs.IT math.IT

Youfa Li , Wenchang Sun This is my paper · ORCID
classification cs.ITmath.IT MSC 94A1242C15
keywords phaselesssamplingphaseretrievalshift-invariantspacegeneralizedHaarconditionrandomdensitycausalsignalschirpzerodistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that random phaseless sampling—recovering a signal from magnitude-only measurements—is possible for causal signals in shift-invariant spaces generated by complex-valued functions, provided the generator satisfies a generalized Haar condition. It first shows that the full spark property, which permits phaseless sampling for real-valued generators, is not enough when the generator is multiplied by a linear phase: such signals are never uniquely recoverable from magnitudes. Replacing the cardinality view of zero sets with a measure-zero condition, the paper proves that three independent uniform samples per unit interval determine any nonseparable causal signal in a complex-generated space up to a global unimodular constant; the real-valued analogue needs only two samples per unit interval. The proof is constructive, giving an alternating phase-decoding and coefficient-recovery algorithm, and numerical experiments recover highly oscillatory chirp signals from a few random samples where deterministic methods would need hundreds.

What carries the argument

The generalized Haar condition (GHC) is the load-bearing object. For $\varphi=\varphi_{\Re}+i\varphi_{\Im}$, the $2s-1$ functions in $\Xi_\varphi=\{\varphi_{\Re}\varphi_{\Re}(\cdot+k)+\varphi_{\Im}\varphi_{\Im}(\cdot+k),\ \varphi_{\Re}\varphi_{\Im}(\cdot+k)-\varphi_{\Im}\varphi_{\Re}(\cdot+k)\}_{k=1}^{s-1}\cup\{\varphi_{\Re}^2+\varphi_{\Im}^2\}$ must be linearly independent, and every nonzero real-linear combination must have a zero set of Lebesgue measure zero on $(0,1)$. This measure-zero property ensures that an auxiliary quantity $A_{n,f}(x,y)+iB_{n,f}(x,y)$ is nonzero with probability 1 and that the two quadratic equations in (2.21) share exactly one solution. The reconstruction then alternates between phase decoding, where the roots of a quadratic provide candidate phases for $f(n+t_{n1})$, and coefficient recovery, where the decoded phase gives $c_n$; the recursion over $n$ advances one unit interval at a time.

What would settle it

Evaluate the determinant in Eq. (1.11) symbolically for the chirp generator family used in the paper: if there are parameter values for which the determinant is identically zero as a function of the sampled points, GHC fails and the density-3 guarantee does not apply to that family. Alternatively, exhibit a nonzero real-linear combination of the functions in $\Xi_\varphi$ whose zero set on $(0,1)$ has positive Lebesgue measure.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.5: if $\varphi$ is a complex-valued GHC-generator with support in $(0,s)$, then every nonseparable causal signal $f=\sum_{k=0}^\infty c_k\varphi(\cdot-k)$ in $V_{\mathrm{ca}}(\varphi)$ is determined up to a unimodular scalar with probability 1 by the random magnitude samples $\{|f(t_0)|\}\cup\{|f(n+t_{n1})|,|f(n+t_{n2})|,|f(n+t_{n3})|: n=1,\dots,\infty\}$, where all offsets are i.i.d. uniform on $(0,1)$. A signal is nonseparable when it cannot be written as a sum of two nonzero elements of the space with disjoint supports; separability is the only obstruction, since separated parts can be rephased independently. The same argument yields Theorem 3.1 for real-valued generators and real-valued signals, where density 2 suffices and the reconstruction is up to a sign. Because the phase decoding is recursive, the restriction of a signal to $[0,L]$ is recovered with probability 1 from finitely many samples, and the number of samples does not grow with the oscillation rate.

Load-bearing premise

The load-bearing premise is the generalized Haar condition: every nonzero real-linear combination of the generator's quadratic products must vanish only on a set of measure zero on $(0,1)$, and for the chirp generators this is verified numerically rather than by a closed-form proof.

Editorial extensions

If this is right

  • In any complex-generated shift-invariant space whose generator satisfies GHC, three random samples per unit interval suffice to recover a nonseparable causal signal up to one global constant of modulus 1.
  • Local reconstruction is cheap: the restriction of such a signal to $[0,L]$ is recovered with probability 1 from $3L-2$ random samples, a count independent of the signal's oscillation rate.
  • For real-valued generators and real-valued signals, the density drops to two random samples per unit interval, with recovery up to a sign.
  • The recursive nature of the algorithm means no global search over phases or coefficients is needed; each step solves a fixed small quadratic system.
  • The result excludes the conjugation ambiguity that plagues conjugate phase retrieval: a complex signal and its conjugate cannot both be valid reconstructions unless the signal is real up to a global phase.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The GHC condition is likely to hold generically among analytic generator families: for generators whose quadratic products are real-analytic on $(0,1)$, GHC reduces to linear independence of those products, since a nonzero analytic function cannot vanish on a positive-measure set. The paper checks this numerically for its chirp generators; a symbolic proof for a parameter family would be a natural
  • A plausible extension is multi-generator shift-invariant spaces: if the analogue of $\Xi_\varphi$ for several generators has the same zero-measure property, the phase-decoding recursion may carry over with a higher sampling density.
  • The noise experiments suggest a qualitative trade-off: binary-phase (real-valued) signals tolerate much lower SNR because the phase decision is a sign decision, while rapidly rotating phases accumulate errors; one testable design rule is to oversample randomly in intervals where the phase function changes quickly.
  • The paper does not address whether density 3 is minimal for complex generators; its impossibility result for linear-phase modulations hints that fewer samples may fail, but the minimum is open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies phaseless sampling (PLS) of causal signals in shift-invariant spaces (SISs) generated by complex-valued generators. It first claims an impossibility theorem (Theorem 1.1) for PLS in SISs generated by a real-valued full-spark function multiplied by a linear phase, then introduces a generalized Haar condition (GHC) on a system of quadratic products of the generator's real and imaginary parts. Under GHC, the paper proves that with probability 1, random sampling density 3 suffices to determine any nonseparable causal signal in a complex-generated SIS up to a unimodular scalar (Theorem 2.5), with density 2 in the real-valued case (Theorem 3.1). The proofs are constructive and lead to recursive phase-decoding/coefficient-recovery algorithms (Approaches II-B and III-B), including local reconstruction guarantees (Propositions 2.6 and 3.2). Numerical simulations for chirp-modulated generators are reported, along with noise-robustness tables.

Significance. The central conditional theorems are potentially significant. The zero-distribution/GHC perspective is a genuinely new tool for phaseless sampling, and the paper gives the first sampling-density guarantees of this type for complex-generated SISs. The constructive nature of the reconstruction algorithms is also a strength, as is the explicit formulation of the GHC hypothesis rather than a fitted assumption. However, the significance is tempered by three gaps: the proof of Theorem 1.1 is invalid as written; the existence of complex GHC-generators is supported only by a floating-point numerical check; and Proposition 2.1, which is load-bearing for the probability-one zero-set lemmas, is asserted without proof. If these gaps are fixed, the paper would be a solid contribution; in its current form the complex-domain claims rest on an unverified premise.

major comments (3)
  1. [§I-B, Theorem 1.1, Eq. (1.3)] The identity (1.3) is false in general. For φ(x)=e^{iαx}ϕ(x), the two sums are e^{iαx}[c0ϕ(x)+c1e^{-iα}ϕ(x-1)] and e^{iαx}[c0ϕ(x)+c1e^{iα}ϕ(x-1)]. Their moduli are |c0ϕ(x)+c1e^{-iα}ϕ(x-1)| and |c0ϕ(x)+c1e^{iα}ϕ(x-1)|, which are not equal for arbitrary c0,c1. For instance, with s=2, N=1, c0=1, c1=e^{iβ}, α=π/4, β=π/3 and ϕ(x)=ϕ(x-1)=1, the two moduli are |1+e^{i(β-α)}| and |1+e^{i(β+α)}|, which differ. Thus the proof of Theorem 1.1 does not establish the claimed impossibility. Since this theorem is a principal motivation for the complex-valued part of the paper, a corrected proof (or a corrected statement) is needed.
  2. [§I-C1, Eq. (1.11) and the chirp example (1.10)] The assertion that φ in (1.10) is a complex-valued GHC-generator is not proved. The paper states that a uniform random search 'found' a nonzero determinant in (1.11), and concludes that the analytic components are linearly independent and hence GHC holds. A finite-precision floating-point determinant value is not a rigorous certificate of linear independence, and the phrase 'with probability 1' is an empirical observation, not a mathematical proof. Because Theorems 2.5 and the related propositions apply only to GHC-generators, the complex-domain contribution currently has no rigorously verified nonempty example class. Please provide either an exact/interval-arithmetic verification for a concrete parameter tuple, or a rigorous proof for a family of generators, or explicitly state that the complex examples are conditional on a numerical conjecture.
  3. [§II-A, Proposition 2.1] Proposition 2.1 is load-bearing but its proof is omitted with the phrase 'can be easily concluded.' The implication is not immediate: GHC is defined for the real span of the quadratic system Ξ_φ, whereas Λ_{φ,1} and Λ_{φ,2} are systems of complex-valued functions, and the zero-measure property must be transferred to their complex spans. This proposition is used in Lemmas 2.10 and 2.11 and in the proof of Theorem 2.5, so a complete proof should be supplied. A natural route is to show that for h in the span of Λ_{φ,1} (or Λ_{φ,2}), the function |h|^2, restricted to (0,1), is a real linear combination of elements of Ξ_φ, so that GHC on Ξ_φ controls the zero set of h.
minor comments (4)
  1. [§III-C, text after Eq. (3.48)] The comparison paragraph says 'section III-C (complex-valued case)' when it appears to refer to the complex-valued simulation in Section II-H; please correct the cross-reference.
  2. [Affiliation, first page] The affiliation string contains a typo: 'Naning' should be 'Nanning'.
  3. [§II-B, Eq. (2.31)] In the displayed formula for step 2 of Approach II-B, the notation zn,k and zn,l is introduced but the argmin expression is a little hard to parse; consider rewriting the tie-breaking/selection rule with explicit definitions of the four candidate roots.
  4. [§V-B, proof of Lemma 2.11] The proof argues separately that the real and imaginary parts of an,f+ibn,f are nonzero with probability 1, then concludes the phase is not jπ/2. This is correct once Proposition 2.1 is available, but the dependence on Proposition 2.1 should be stated explicitly in this lemma, since it is not immediately visible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sampling-density theorems are conditional on the explicit GHC hypothesis, and the reconstruction recurses from samples; the numerical GHC check is a verification gap, not a circular reduction.

full rationale

The paper's main theorems (2.5 and 3.1) are conditional on the generalized Haar condition (GHC), which is an explicit hypothesis on the generator (Eqs. 1.8 and 1.9), not a parameter fitted to data. The reconstruction in Approach II-B is a recursive coefficient recovery from magnitude samples using the known generator; it never uses the target signal to determine constants beyond the unavoidable global phase. The probability-one statements are derived in Appendix V from GHC and analytic zero-set properties, not from the conclusion being assumed. Theorem 1.1 is a negative counterexample, not a circular input. The only notable weakness is the numerical determinant check in Eq. (1.11) for the chirp generator, which is reported as a 'found' nonzero determinant without rigorous error bounds; this is a validation gap rather than a circular reduction, and the theorems' content is unaffected because GHC is stated as a hypothesis. Self-citations to [23], [24], and [48] are background examples and are not load-bearing in the proofs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces a new technical condition (GHC) and assumes it for the generator in all main theorems. It also assumes causal, nonseparable signals and compact support. No numerical constants are fitted to data; the sampling densities are derived from GHC. No new physical entities are postulated.

assumptions (7)
  • domain assumption The generator φ satisfies GHC (1.8): the real span of Ξφ has every nonzero element with zero set of Lebesgue measure zero on (0,1).
    Assumed in Theorems 2.5, 2.6 and in Lemmas 2.10-2.12; the paper only verifies this numerically for the chirp examples in Sec I-C1.
  • domain assumption The real generator ϕ satisfies GHC (1.9): every nonzero element of span{Λϕ} has zero set of measure zero on (0,1).
    Assumed in Theorem 3.1 and Proposition 3.2; B-splines and refinable functions with positive masks are cited as examples.
  • domain assumption Signals are causal and nonseparable, with c0 ≠ 0.
    Nonseparability is necessary because separable signals are indistinguishable by construction (f vs f1 + e^{iα}f2); used to obtain G_f < s-1 in Lemma 2.9.
  • domain assumption The generator has compact support supp(φ) ⊆ (0,s) or supp(ϕ) ⊆ (0,s), integer s ≥ 2.
    Used throughout to make v_{n,f} a finite sum and to define the index sets I_n in (2.12).
  • domain assumption Random sampling points are i.i.d. uniform on (0,1).
    The probability-1 statements are with respect to this distribution.
  • standard math Nonzero analytic functions have zero sets of Lebesgue measure zero.
    Used in Sec I-C to verify GHC for analytic chirp generators; standard result cited as [45].
  • domain assumption Full spark property of the matrix (1.2) for real ϕ in Theorem 1.1.
    Assumed to hold for the motivating impossibility result.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective." pith.science (2026). https://pith.science/paper/GQ6ET6HW

@misc{pith2026190805423,
  author       = {Pith},
  title        = {Pith review of: Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQ6ET6HW}},
  note         = {Machine review of arXiv:1908.05423}
}
abstract

We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{\textbf{i}\alpha \cdot}\varphi), \alpha\neq0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{\textbf{i}\alpha \cdot}\varphi$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability $1$, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability $1$, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 47 canonical work pages

  1. [1]

    R. W. Gerchberg and W. O. Saxton, A practical algorithm fo r the determination of phase from image and diffraction plane pic tures, Optik, 35, 237-246, 1972

    work page 1972

  2. [2]

    J. R. Fienup, Phase retrieval algorithms: A comparison, Applied Optics, 21(15), 2758-2769, 1982

    work page 1982

  3. [3]

    J. R. Fienup, Reconstruction of an object from the modulu s of its Fourier transform, Optics Letters , 3(1), 27-29, 1978

    work page 1978

  4. [4]

    J. R. Fienup, Phase retrieval algorithms: A personal tou r, Applied Optics, 52(1), 45-56, 2013

    work page 2013

  5. [5]

    E. J. Cand` es, T. Strohmer and V . V oroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via con vex programming, Communications on Pure and Applied Mathematics , 66(8), 1241-1274, 2013

    work page 2013

  6. [6]

    B. A. Shenoy and C. S. Seelamantula, Exact phase retrieva l for a class of 2-D parametric signals, IEEE Transactions on Signal Processing, 63(1), 90-103, 2015

    work page 2015

  7. [7]

    Shechtman, Y

    Y . Shechtman, Y . C. Eldar, O. Cohen, H. N. Chapman, J. Miao and M. Segev, Phase retrieval with application to optical im aging, IEEE Signal Processing Magazine , 32(3), 87-109, 2015

    work page 2015

  8. [8]

    J. Miao, T. Ishikawa, I.K. Robinson and M.M. Murnane, Bey ond crystallography: Diffractive imaging using coherent x-ra y light sources, Science, 348(6234), 530-535, 2015

    work page 2015

Show all 48 references
  1. [9]

    E. J. Cand` es, Y . C. Eldar, T. Strohmer and V . V oroninski, Phase retrieval via matrix completion, SIAM Journal on Imaging Sciences, 6(1), 199-225, 2013

    2013

  2. [10]

    Heinosaarri, L

    T. Heinosaarri, L. Mazzarella and M. M.Wolf, Quantum to mog- raphy under prior information, Communications in Mathematical Physics, 318, 355-374, 2013

    2013

  3. [11]

    Balan, P .G

    R. Balan, P .G. Casazza and D. Edidin, On signal reconstr uction without noisy phase, Applied and Computational Harmonic Anal- ysis, 20, 345-356, 2006

    2006

  4. [12]

    L. Li, C. Cheng, D. Han, Q. Sun and G. Shi, Phase retrieval from multiple-window short-time Fourier measurements, IEEE Signal Processing Letters, 24(4), 372-376, 2017

    2017

  5. [13]

    Huang, Y

    K. Huang, Y . C. Eldar and N. D. Sidiropoulos, Phase retri eval from 1D Fourier measurements: convexity, uniqueness, and algori thms, IEEE Transactions on Signal Processing, 64(23), 6105-6117, 2016

    2016

  6. [14]

    Y . Chen, C. Cheng and Q. Sun, Phase retrieval of complex a nd vector-valued functions, arXiv preprint arXiv: 1909.02078v1

    1909 arXiv

  7. [15]

    Alaifari, I

    R. Alaifari, I. Daubechies, P . Grohs and G. Thakur, Reco nstructing real-valued functions from unsigned coeffcients with resp ect to wavelet and other frames, Journal of F ourier Analysis and Ap- plications, 23, 1480-1494, 2017

    2017

  8. [16]

    Alaifari, I

    R. Alaifari, I. Daubechies, P . Grohs and R. Yin, Stable p hase retrieval in infinite dimensions, F oundations of Computational Mathematics, 19, 869-900, 2019

    2019

  9. [17]

    Cahill, P

    J. Cahill, P . G. Casazza and I. Daubechies, Phase retrie val in infinite-dimensional Hilbert spaces, Trans. Amer . Math. Soc. Ser . B, 3, 63-76, 2016

    2016

  10. [18]

    Shlezinger, R

    N. Shlezinger, R. Dabora and Y . C. Eldar, Measurement ma trix design for phase retrieval based on mutual information, IEEE Transactions on Signal Processing, 66(2), 324-339, 2018

    2018

  11. [19]

    B. A. Shenoy, S. Mulleti and C. S. Seelamantula, Exact ph ase retrieval in principal shift-invariant spaces, IEEE Transactions on Signal Processing, 64(2), 406-416, 2016

    2016

  12. [20]

    Aldroubi and K

    A. Aldroubi and K. Gr¨ ochenig, Nonuniform sampling and recon- struction in shift-invariant spaces, SIAM Rev., 43, 585-620, 2001

    2001

  13. [21]

    J. Shi, X. Liu, F. G Y an and W. Song, Error analysis of reco n- struction from linear canonical transform based sampling, IEEE Transactions on Signal Processing, 66(7), 1748-1760, 2018

    2018

  14. [22]

    Bhandari and A

    A. Bhandari and A. I. Zayed, Shift-Invariant and sampli ng spaces associated with the fractional Fourier transform domain, IEEE Transactions on Signal Processing, 60(4), 1627-1637, 2012

    2012

  15. [23]

    Sun, Phaseless sampling and linear reconstruction o f functions in spline spaces, arXiv preprint, arXiv:1709.04779

    W. Sun, Phaseless sampling and linear reconstruction o f functions in spline spaces, arXiv preprint, arXiv:1709.04779

    arXiv

  16. [24]

    Sun, Local and global phaseless sampling in real spli ne spaces, arXiv preprint, arXiv:1705.00836

    W. Sun, Local and global phaseless sampling in real spli ne spaces, arXiv preprint, arXiv:1705.00836. 16

    arXiv

  17. [25]

    Thakur, Reconstruction of bandlimited functions fr om unsigned samples, Journal of F ourier Analysis and Applications, 17(4), 720- 732, 2011

    G. Thakur, Reconstruction of bandlimited functions fr om unsigned samples, Journal of F ourier Analysis and Applications, 17(4), 720- 732, 2011

    2011

  18. [26]

    Y . Chen, C. Cheng, Q. Sun and H. Wang, Phase retrieval of r eal- valued signals in a shift-invariant space, Applied and Computational Harmonic Analysis , 49, 56-73, 2020

    2020

  19. [27]

    Cheng, J

    C. Cheng, J. Jiang and Q. Sun, Phaseless sampling and rec on- struction of real-valued signals in shift-invariant space s, Journal of F ourier Analysis and Applications, 25, 1361-1394, 2019

    2019

  20. [28]

    Jaming, K

    P . Jaming, K. Kellay and R. Perez Iii, Phase retrieval fo r wide-band signals, arXiv preprint arXiv:1905.04095

    1905 arXiv

  21. [29]

    C.K. Lai, F. Littmann and E. Weber, Conjugate phase retr ieval in Paley-Wiener space, arXiv preprint arXiv: 1910.12975

    1910 arXiv

  22. [30]

    Selva, Interpolation of bounded bandlimited signal s and applica- tions, IEEE Transactions on Signal Processing, 54(11), 4244-4260, 2006

    J. Selva, Interpolation of bounded bandlimited signal s and applica- tions, IEEE Transactions on Signal Processing, 54(11), 4244-4260, 2006

    2006

  23. [31]

    Han, Framelets and wavelets: Algorithms, analysis, and applications, Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, Cham, 2017

    B. Han, Framelets and wavelets: Algorithms, analysis, and applications, Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, Cham, 2017. xxxiii +724 pp

    2017

  24. [32]

    Marks, J

    R. Marks, J. Walkup and M. Hagler, Sampling theorems for linear shift-variant systems, IEEE Transactions on Circuits and Systems , 25(4), 228-233, 1978

    1978

  25. [33]

    Meinsma and L

    G. Meinsma and L. Mirkin, L2 Sampled signal reconstruction with causality constraints - Part I: Setup and solutions, IEEE Transactions on Signal Processing , 60(5), 2260-2272, 2012

    2012

  26. [34]

    Bhandari and A

    A. Bhandari and A. I. Zayed, Shift-invariant and sampli ng spaces associated with the special affine Fourier transform, Applied and Computational Harmonic Analysis , 47(1), 30-52, 2019

    2019

  27. [35]

    Han, Frame representations and Parseval duals with a pplications to Gabor frames, Transactions of the American Mathematical Society, 360(6), 3307-3326, 2008

    D. Han, Frame representations and Parseval duals with a pplications to Gabor frames, Transactions of the American Mathematical Society, 360(6), 3307-3326, 2008

    2008

  28. [36]

    Han and D

    D. Han and D. Larson, Frames, bases and group representa tions, Memoirs of the American Mathematical Society, 697, 2000

    2000

  29. [37]

    C. K. Chui, P . W. Swih and J. D. Ward, Best L2 local approxi- mation, Journal of Approximation Theory , 22, 254-261, 1978

    1978

  30. [38]

    B Dunham, Families satisfying the Haar condition, Journal of Approximation Theory, 12, 291-298, 1974

    C. B Dunham, Families satisfying the Haar condition, Journal of Approximation Theory, 12, 291-298, 1974

    1974

  31. [39]

    V an de V el, The Haar condition and multiplicity of zer os, Numerische Mathematik, 39, 139-153, 1982

    H. V an de V el, The Haar condition and multiplicity of zer os, Numerische Mathematik, 39, 139-153, 1982

    1982

  32. [40]

    E. J. Cand `es, P . R. Charlton and H. Helgason, Detecting highly oscillatory signals by chirplet path pursuit, Applied and Computa- tional Harmonic Analysis, 24(1), 14-40, 2008

    2008

  33. [41]

    Simmons, Echolocation in bats: Signal processing of echoes for target range, Science, 171 (974), 925-928, 1971

    J. Simmons, Echolocation in bats: Signal processing of echoes for target range, Science, 171 (974), 925-928, 1971

    1971

  34. [42]

    Reynolds III and S.A

    J.E. Reynolds III and S.A. Rommel, Biology of Marine Mam mals, Smithsonian Institution Press, W ashington, DC, 1999

    1999

  35. [43]

    X. Gu, S. Akturk and R. Trebino, Spatial chirp in ultrafa st optics, Optics Communications, 242(4), 599-604, 2004

    2004

  36. [44]

    Akturk, X

    S. Akturk, X. Gu, P . Bowlan and R. Trebino, Spatio-tempo ral couplings in ultrashort laser pulses, Journal of Optics, 12, 093001, 2010

    2010

  37. [45]

    Garnett, Bounded analytic functions, Graduate Texts in Mathe- matics, 236, Springer, 2007

    J. Garnett, Bounded analytic functions, Graduate Texts in Mathe- matics, 236, Springer, 2007

    2007

  38. [46]

    T. N. T. Goodman and C. A. Micchelli, On refinement equati ons determined by P´ olya frequency sequences,SIAM Journal on Math- ematical Analysis , 23, 766-784, 1992

    1992

  39. [47]

    T. N. T. Goodman and Q. Sun, Total positivity and refinabl e func- tions with general dilation, Applied and Computational Harmonic Analysis, 16, 69-89, 2004

    2004

  40. [48]

    Y ang and Y

    S. Y ang and Y . Li, Two-direction refinable functions and two- direction wavelets with high approximation order and regul arity, Science in China Series A: Mathematics , 50, 1687-1704, 2007

    2007

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.