School of Mathematics
Georgia Institute of Technology
Atlanta, GA 303320160
(404) 8949231
h e i l @ m a t h . g a t e c h . e d u
Links to many individual research papers are provided below.
Please email me to request a copy of any paper that is not available
electronically.
Categories:
Recent Papers
The HRT Conecture
Surveys and Pretty Good Expository Papers
Localized Frames, Density, Excess, and the HAP
Pseudodifferential Operators
TimeFrequency Analysis and Gabor Systems
Frame Theory, Sampling,
and ShiftInvariant Spaces
Wavelets, Refinable Functions,
and the Joint Spectral Radius
Image Processing
Generalized Harmonic Analysis
Conference Proceedings and Other Publications
Book Reviews
Recent Papers
 L. Hart, C. Heil, I. Katz, and M. Northington V,
Overcomplete reproducing pairs, arXiv:2311.04421.
 C. Heil and P.T. Yu,
ℓ^{1}bounded sets, arXiv:2307.05536.
 C. Heil and P.T. Yu,
Convergence of frame series,
J. Fourier Anal. Appl., 29 (2023), no. 1, Paper No. 14, 13 pages.
 Y.S. Cheng and C. Heil,
Existence of finite unitnorm tight frames in Banach spaces,
Graduate J. Math., 7 (2022), 1738.
 C. Heil,
Absolute Continuity and the BanachZaretsky Theorem,
in: "Excursions in Harmonic Analysis," Volume 6
M. Hirn et al., eds., Birkhäuser, Cham (2021), 2751.
Related lecture on
Absolute Continuity and the BanachZaretsky Theorem,
presented at the
Faraway Fourier Talks on March 29, 2021.
This talk gives a streamlined discussion of the main topics from
Chapters 5 and 6 of my text "Introduction to Real Analysis".
The HRT Conjecture
 C. Heil and D. Speegle,
The HRT Conjecture and the Zero Divisor Conjecture for the
Heisenberg group,
in: "Excursions in Harmonic Analysis," Volume 3,
R. Balan et al., eds., Birkhäuser/Springer, Cham (2015), 159176.
 C. Heil,
Linear independence of finite Gabor systems,
in: "Harmonic Analysis and Applications,"
Birkhäuser, Boston (2006), 171206.
Errata.
 C. Heil, J. Ramanathan, and P. Topiwala,
Linear independence of timefrequency translates,
Proc. Amer. Math. Soc., 124 (1996), 27872795.
Surveys and Pretty Good Expository Papers
 C. Heil,
Absolute Continuity and the BanachZaretsky Theorem,
in: "Excursions in Harmonic Analysis," Volume 6
M. Hirn et al., eds., Birkhäuser, Cham (2021), 2751.
Related lecture on
Absolute Continuity and the BanachZaretsky Theorem,
presented at the
Faraway Fourier Talks on March 29, 2021.
This talk gives a streamlined discussion of the main topics from
Chapters 5 and 6 of my text "Introduction to Real Analysis".
 C. Heil,
A Brief Guide to Metrics, Norms, and Inner Products,
2016 (electronic manuscript, 66 pages).
A greatly expanded version of this manuscript has been
published by Birkhäuser under the title
"Metrics, Norms, Inner Products, and Operator Theory."
 C. Heil,
WHAT IS a Frame?,
Notices Amer. Math. Soc., 60 (2013), 748750.
Copyright 2013 by the American Mathematical Society.
 C. Heil,
The Density Theorem and the Homogeneous Approximation Property
for Gabor frames, in: "Representations, Wavelets, and Frames:
A Celebration of the Mathematical Work of Lawrence Baggett,"
P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, eds.,
Birkhäuser, Boston (2008), 71102.
 C. Heil and D. R. Larson,
Operator theory and modulation spaces,
in: "Frames and Operator Theory in Analysis and Signal Processing"
(San Antonio, 2006),
Contemp. Math., Vol. 451, Amer. Math. Soc., Providence, RI (2008), 137150.
 C. Heil,
History and evolution of the Density Theorem for Gabor frames,
J. Fourier Anal. Appl., 13 (2007), 113166.
 C. Heil,
Integral operators, pseudodifferential operators, and Gabor frames,
in: "Advances
in Gabor Analysis," H. G. Feichtinger and T. Strohmer, eds.,
Birkhäuser, Boston (2003), 153169.
 C. Heil,
An introduction to weighted Wiener amalgams,
in: "Wavelets and their Applications" (Chennai, January 2002),
M. Krishna, R. Radha and S. Thangavelu, eds.,
Allied Publishers, New Delhi (2003), 183216.
 C. Heil,
A Basis Theory Primer, 1998 (electronic manuscript, 93 pages).
Note: This is the original very short set of notes; the much larger
and vastly improved
Expanded Edition is available from Birkhäuser!
 J. J. Benedetto, C. Heil, and D. F. Walnut,
Differentiation and the BalianLow theorem,
J. Fourier Anal. Appl., 1 (1995), 355402.
 C. Heil and G. Strang,
Continuity of the joint spectral radius: Application to wavelets,
in: "Linear Algebra for Signal Processing" (Minneapolis, MN, 1992),
A. Bojanczyk and G. Cybenko, eds., IMA Vol. Math. Appl. 69,
SpringerVerlag, New York (1995), 5161.
 D. Colella and C. Heil,
Dilation equations and the smoothness of compactly supported wavelets,
in: "Wavelets: Mathematics and Applications,"
J. J. Benedetto and M. W. Frazier, eds.,
CRC Press, Boca Raton, FL (1994), 163201.
 C. Heil,
Methods of solving dilation equations,
in: "Probabilistic and Stochastic Methods in Analysis, with Applications"
(Il Ciocco, 1991), J. S. Byrnes et al., eds., NATO Adv. Sci. Inst. Ser. C:
Math. Phys. Sci. 372, Kluwer, Dordrecht (1992), 1545.
 C. E. Heil and D. F. Walnut,
Continuous and discrete wavelet transforms,
SIAM Review, 31 (1989), 628666.
Localized Frames, Density, Excess, and the HAP
 C. Heil and G. Kutyniok,
Density of frames and Schauder bases of windowed exponentials,
Houston J. Math., 34 (2008), 565600.
 C. Heil,
The Density Theorem and the Homogeneous Approximation Property
for Gabor frames, in: "Representations, Wavelets, and Frames:
A Celebration of the Mathematical Work of Lawrence Baggett,"
P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, eds.,
Birkhäuser, Boston (2008), 71102.
 C. Heil and G. Kutyniok,
The Homogeneous Approximation Property for wavelet frames,
J. Approx. Theory, 147 (2007), 2846.
 C. Heil,
History and evolution of the Density Theorem for Gabor frames,
J. Fourier Anal. Appl., 13 (2007), 113166.
 R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Density, overcompleteness, and localization of frames,
Electron. Res. Announc. Amer. Math. Soc., 12 (2006), 7186.
 R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Density, overcompleteness, and localization of frames, II. Gabor systems,
J. Fourier Anal. Appl., 12 (2006), 307344.
Errata.
 R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Density, overcompleteness, and localization of frames, I. Theory,
J. Fourier Anal. Appl., 12 (2006), 105143.
Transparencies from a related
talk given at the
2nd International Conference on Computational Harmonic Analysis,
Vanderbilt University, May 27, 2004, and another
talk given at the University of Maryland, March 14, 2005.
 C. Heil and G. Kutyniok,
Density of weighted wavelet frames,
J. Geometric Analysis, 13 (2003), pp. 479493.
 R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Excesses of Gabor frames,
Appl. Comput. Harmon. Anal., 14 (2003), 87106.
 R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Deficits and excesses of frames,
Adv. Comput. Math.,
Special Issue on Frames, 18 (2003), 93116.
 B. Deng and C. Heil,
Density of Gabor Schauder bases,
in: "Wavelet Applications in Signal and Image Processing VIII,"
(San Diego, CA, 2000), Proc. SPIE Vol. 4119, A. Aldroubi et al., eds.,
SPIE, Bellingham, WA (2000), 153164.
 O. Christensen, B. Deng, and C. Heil,
Density of Gabor frames,
Appl. Comput. Harmon. Anal., 7 (1999), 292304.
Errata.
Pseudodifferential Operators
 Á. Bényi, K. Gröchenig, C. Heil, and K. Okoudjou,
Modulation spaces and a class of bounded multilinear pseudodifferential
operators, J. Operator Theory, 54 (2005), 387399.
 K. Gröchenig and C. Heil,
Counterexamples for boundedness of pseudodifferential operators,
Osaka J. Math., 41 (2004), 681691.
 C. Heil,
Integral operators, pseudodifferential operators, and Gabor frames,
in: "Advances
in Gabor Analysis," H. G. Feichtinger and T. Strohmer, eds.,
Birkhäuser, Boston (2003), 153169.
 K. Gröchenig and C. Heil,
Modulation spaces as symbol classes for pseudodifferential operators,
in: "Wavelets and their Applications" (Chennai, January 2002),
M. Krishna, R. Radha and S. Thangavelu, eds.,
Allied Publishers, New Delhi (2003), 151169.
 K. Gröchenig and C. Heil,
Modulation spaces and pseudodifferential operators,
Integral Equations Operator Theory, 34 (1999), 439457.
 C. Heil, J. Ramanathan, and P. Topiwala,
Singular values of compact pseudodifferential operators,
J. Funct. Anal., 150 (1997), 426452.
Errata.
TimeFrequency Analysis and Gabor Systems
 R. Tinaztepe and C. Heil,
Modulation spaces, BMO, and the BalianLow Theorem,
Sampl. Theory Signal Image Process., 11 (2012), 2541.
 C. Heil and A. M. Powell,
Regularity for complete and minimal Gabor systems on a lattice,
Illinois J. Math., 53 (2010), 10771094.
 C. Heil and D. R. Larson,
Operator theory and modulation spaces,
in: "Frames and Operator Theory in Analysis and Signal Processing"
(San Antonio, 2006),
Comtemp. Math., Vol. 451, Amer. Math. Soc., Providence, RI (2008), 137150.
 C. Heil and A. M. Powell,
Gabor Schauder Bases and the BalianLow Theorem,
J. Math. Physics, 47 (2006).
 C. Heil,
Linear independence of finite Gabor systems,
in: "Harmonic Analysis and Applications,"
Birkhäuser, Boston (2006), 171206.
Errata.
 K. Gröchenig, C. Heil, and K. Okoudjou,
Gabor analysis in weighted amalgam spaces,
Sampling Theory in Signal and Image Processing, 1 (2003), 225259.
 K. Gröchenig, D. Han, C. Heil, G. Kutyniok,
The BalianLow theorem for symplectic lattices in higher dimensions,
Appl. Comput. Harmon. Anal., 13 (2002), 169176.
 K. Gröchenig and C. Heil,
Gabor meets LittlewoodPaley: Gabor expansions in L^p(R^d),
Studia Math., 146 (2001), 1533.
 J. J. Benedetto, C. Heil, and D. F. Walnut,
Gabor systems and the BalianLow theorem, in:
"Gabor Analysis and Algorithms: Theory and Applications,"
H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston (1998), 85122.
 J. J. Benedetto, C. Heil, and D. F. Walnut,
Differentiation and the BalianLow theorem,
J. Fourier Anal. Appl., 1 (1995), 355402.
 J. Benedetto, C. Heil, and D. Walnut,
Uncertainty Principles for timefrequency operators,
in: "Continuous and Discrete Fourier Transforms, Extension Problems and
WienerHopf Equations," Oper. Theory Adv. Appl. 58,
I. Gohberg, ed., Birkhäuser, Basel (1992), 125.
 C. Heil and D. Walnut,
Gabor and wavelet expansions,
in: "Recent Advances in Fourier Analysis and its Applications"
(Il Ciocco, 1989), J. S. Byrnes and J. L. Byrnes, eds., NATO Adv. Sci. Inst.
Ser. C: Math. Phys. Sci. 315, Kluwer, Dordrecht (1990), 441454.
 C. Heil,
A discrete Zak transform,
Technical Report, The MITRE Corporation, December 1989.
 C. E. Heil and D. F. Walnut,
Continuous and discrete wavelet transforms,
SIAM Review, 31 (1989), 628666.
Frame Theory, Sampling, and ShiftInvariant Spaces
 L. Hart, C. Heil, I. Katz, and M. Northington V,
Overcomplete reproducing pairs, arXiv:2311.04421.
 C. Heil and P.T. Yu,
ℓ^{1}bounded sets, arXiv:2307.05536.
 C. Heil and P.T. Yu,
Convergence of frame series,
J. Fourier Anal. Appl., 29 (2023), no. 1, Paper No. 14, 13 pages.
 Y.S. Cheng and C. Heil,
Existence of finite unitnorm tight frames in Banach spaces,
Graduate J. Math., 7 (2022), 1738.
 G. J. Yoon and C. Heil,
Duals of windowed exponential systems,
Acta Appl. Math., 119 (2012), 97112.
The
published version is available at www.springerlink.com.
Here is a related lecture
from the
Codex seminar series.
 A. Aldroubi, C. Cabrelli, C. Heil, K. Kornelson, and U. Molter,
Invariance of a shiftinvariant space,
J. Fourier Anal. Appl., 16 (2012), 6075.
 S. Bishop, C. Heil, Y. Y. Koo, and J. K. Lim,
Invariances of frame sequences under perturbation,
Linear Algebra Appl., 432 (2010), 15011514.
 C. Heil, Y. Y. Koo, and J. K. Lim,
Duals of frame sequences, Acta Appl. Math., 107 (2009), 7590.
The
published version is available at www.springerlink.com.
 K. Gröchenig, C. Heil, and D. Walnut,
Nonperiodic sampling and the local three squares theorem,
Ark. Mat., 38 (2000), 7792.
 O. Christensen and C. Heil,
Perturbations of Banach frames and atomic decompositions,
Math. Nachr., 186 (1997), 3347.
 C. E. Heil and D. F. Walnut,
Continuous and discrete wavelet transforms,
SIAM Review, 31 (1989), 628666.
Wavelets, Refinable Functions, and the Joint Spectral Radius
 C. Heil, D. Jacobs, and R. Tinaztepe,
Smoothness of refinable function vectors on R^n,
Int. J. Wavelets Multiresolut. Inf. Process.,
15 (2017) 1750051 (16 pages).
 C. A. Cabrelli, C. Heil, and U. M. Molter,
Selfsimilarity and multiwavelets in higher dimensions,
Memoirs Amer. Math. Soc., Vol. 170, No. 807 (2004), 82 pages.
 C. A. Cabrelli, C. Heil, and U. M. Molter,
Multiwavelets in R^n with an arbitrary dilation matrix,
in: "Wavelets and Signal Processing," L. Debnath, ed.,
Birkhäuser, Boston (2003), 2339.
 C. Cabrelli, C. Heil, and U. Molter,
Accuracy of several multidimensional refinable distributions,
J. Fourier Anal. Appl., 6 (2000), 483502.
 C. A. Cabrelli, C. Heil, and U. M. Molter,
Necessary conditions for the existence of multivariate multiscaling
functions, in: "Wavelet Applications in Signal and Image Processing VIII"
(San Diego, CA, 2000), Proc. SPIE Vol. 4119, A. Aldroubi et al., eds.,
SPIE, Bellingham, WA (2000), 395406.
 C. A. Cabrelli, C. Heil, and U. M. Molter,
Polynomial reproduction by refinable functions,
in: "Advances in Wavelets" (Hong Kong, 1997),
K.S. Lau, ed., SpringerVerlag, Singapore (1999), 121161.
 C. Cabrelli, C. Heil, and U. Molter,
Accuracy of lattice translates of several multidimensional refinable
functions,
J. Approx. Theory, 95 (1998), 552.
Errata.
 C. Heil and D. Colella,
Matrix refinement equations: Existence and uniqueness,
J. Fourier Anal. Appl., 2 (1996), 363377.
 C. Heil, G. Strang and V. Strela,
Approximation by translates of refinable functions,
Numerische Math., 73 (1996), 7594.
Errata.
 C. Heil and G. Strang,
Continuity of the joint spectral radius: Application to wavelets,
in: "Linear Algebra for Signal Processing" (Minneapolis, MN, 1992),
A. Bojanczyk and G. Cybenko, eds., IMA Vol. Math. Appl. 69,
SpringerVerlag, New York (1995), 5161.
 C. Heil and D. Colella,
Sobolev regularity for scaling functions via ergodic theory,
in: "Approximation Theory VIII," Vol. 2 (College Station, TX, 1995),
C. K. Chui and L. L. Schumaker, eds.,
World Scientific, Singapore (1995), 151158.
 D. Colella and C. Heil,
Dilation equations and the smoothness of compactly supported wavelets,
in: "Wavelets: Mathematics and Applications,"
J. J. Benedetto and M. W. Frazier, eds.,
CRC Press, Boca Raton, FL (1994), 163201.
 D. Colella and C. Heil,
Characterizations of scaling functions: Continuous solutions,
SIAM J. Matrix Anal. Appl., 15 (1994), 496518.
 C. Heil,
Some stability properties of wavelets and scaling functions,
in: "Wavelets and Their Applications: (Il Ciocco, 1992),
J. S. Byrnes et al., eds., NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci.
442, Kluwer, Dordrecht (1994), 1938.
 D. Colella and C. Heil,
The characterization of continuous, fourcoefficient scaling functions
and wavelets, IEEE Trans. Inform. Theory, 38 (1992), 876881.
 C. Heil,
Methods of solving dilation equations,
in: "Probabilistic and Stochastic Methods in Analysis, with Applications"
(Il Ciocco, 1991), J. S. Byrnes et al., eds., NATO Adv. Sci. Inst. Ser. C:
Math. Phys. Sci. 372, Kluwer, Dordrecht (1992), 1545.
 C. E. Heil and D. F. Walnut,
Continuous and discrete wavelet transforms,
SIAM Review, 31 (1989), 628666.
Image Processing
 R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt,
Pseudodifferential operators, microlocal analysis and image restoration, in:
"Advances in PseudoDifferential Operators,"
R. Ashino, P. Boggiatto, and M.W. Wong, eds.,
Birkhäuser, Boston (2004), 187202.
 R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt,
Smooth tight frame wavelets and image analysis in Fourier space,
Comput. Math. Appl., 45 (2003), 15511579.
 R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt,
Microlocal analysis, smooth frames and denoising in Fourier space,
Asian InformationScienceLife, 1 (2002), 153160.
 R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt,
Microlocal filtering with multiwavelets,
Comput. Math. Appl., 41 (2001), 111133.
 V. Strela, P. N. Heller, G. Strang, P. Topiwala, and C. Heil,
The application of multiwavelet filterbanks to image processing,
IEEE Trans. Image Proc., 8 (1999), 548563.
Generalized Harmonic Analysis
Conference Proceedings and Other
Publications
 S. Bishop, C. Heil, Y. Y. Koo, and J. K. Lim,
Duals and invariances of frame sequences,
in: "Wavelets XIII" (San Diego, CA, 2009), Proc. SPIE Vol. 7446,
V. Goyal et al., eds., SPIE, Bellingham, WA (2009), 74460K174460K8.
 C. Heil and G. Kutyniok,
Convolution and Wiener amalgam spaces on the affine group,
in: "Recent Advances in Computational Science,"
P. E. T. Jorgensen, X. Shen, C.W. Shu, and N. Yan, eds.,
World Scientific, Singapore (2008), 209217.
 R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Excess of Parseval frames,
in: "Wavelets XI" (San Diego, CA, 2005), Proc. SPIE Vol. 5914,
M. Papadakis et al., eds., SPIE, Bellingham, WA (2005), 3946.
 R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt,
Image restoration through microlocal analysis with smooth tight
wavelet frames, in: "Theoretical development and feasibility of mathematical
analysis on the computer" (Kyoto, 2002), Surikaisekikenkyusho Kokyuroku
No. 1286 (2002), 101118.
 R. Ashino, C. Heil, M. Nagase, and R. Vaillancourt,
Multiwavelets, pseudodifferential operators and microlocal analysis,
in: "Wavelet Analysis and Applications" (Guangzhou, China, 1999),
D. Deng et al., eds., AMS/IP Stud. Adv. Math., 25,
American Mathematical Society, Providence, RI (2002), 920.
 R. Ashino, C. Heil, M. Nagase, and R. Vaillancourt,
Microlocal analysis and multiwavelets, in:
"Geometry, Analysis and Applications" (Varanasi, India, 2000),
R. S. Pathak, ed., World Scientific, Singapore (2001), 293302.
 C. Heil,
Wavelets, Section 7.13.6 in the CRC Standard Mathematical Tables and Formulae,
30th Edition, D. Zwillinger, ed., CRC Press, Boca Raton, FL (1996), 663667
(Section 7.15.5 in the 31st Edition, 2003, 723726).
 C. Heil,
Existence and accuracy for matrix refinement equations,
Z. Angew. Math. Mech., Special issue on Applied Stochastics and Optimization,
76 (1996), 251254.
 P. N. Heller, V. Strela, G. Strang, P. Topiwala,
C. Heil, and L. S. Hills,
Multiwavelet filter banks for data compression,
in: ISCAS '95, Proc. International Symposium on Circuits and Systems
(Seattle, WA, 1995), Vol. 3, IEEE, Piscataway, NJ (1995), 17961799.
 C. Heil, J. Ramanathan, and P. Topiwala,
Asymptotic singular value decay of timefrequency localization operators,
in: "Wavelet Applications in Signal and Image Processing II"
(San Diego, CA, 1994), Proc. SPIE Vol. 2303, A. F. Laine and M. A. Unser, eds.,
SPIE, Bellingham, WA (1994), 1524.
 C. Heil,
Applications of the fast wavelet transform,
in: "Advanced SignalProcessing Algorithms, Architectures, and Implementations"
(San Diego, CA, 1990), Proc. SPIE Vol. 1348, F. T. Luk, ed.,
SPIE, Bellingham, WA (1990), 248259.
 C. Heil,
Wavelets and frames,
in: "Signal Processing, Part I: Signal Processing Theory,"
L. Auslander, T. Kailath, and S. Mitter, eds., IMA Vol. Math. Appl. 22,
SpringerVerlag, New York (1990), 147160.
Book Reviews

Review of "Ten Lectures on Wavelets"
by I. Daubechies (review appeared in SIAM Review, 35 (1993), 666669).

Review of "A First Course in Fourier Analysis,"
by D. W. Kammler (review appeared in SIAM Review, 43 (2001) 722724).