Christopher E. Heil

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:
The HRT Conecture
Surveys and Pretty Good Expository Papers
Localized Frames, Density, Excess, and the HAP
Pseudodifferential Operators
Time-Frequency Analysis and Gabor Systems
Frame Theory, Sampling, and Shift-Invariant Spaces
Wavelets, Refinable Functions, and the Joint Spectral Radius
Image Processing
Generalized Harmonic Analysis
Conference Proceedings and Other Publications
Book Reviews

• 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), 159--176.

• C. Heil, Linear independence of finite Gabor systems, in: "Harmonic Analysis and Applications," Birkhäuser, Boston (2006), 171-206.   Errata.

• C. Heil, J. Ramanathan, and P. Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc., 124 (1996), 2787-2795.

• C. Heil, Absolute Continuity and the Banach-Zaretsky Theorem, in: "Excursions in Harmonic Analysis," Volume 6 M. Hirn et al., eds., Birkhäuser, Cham (2021), 27-51.

• 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), 748-750. 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), 71-102.

• 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), 137-150.

• C. Heil, History and evolution of the Density Theorem for Gabor frames, J. Fourier Anal. Appl., 13 (2007), 113-166.

• C. Heil, Integral operators, pseudodifferential operators, and Gabor frames, in: "Advances in Gabor Analysis," H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston (2003), 153-169.

• 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), 183-216.

• 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 Balian-Low theorem, J. Fourier Anal. Appl., 1 (1995), 355-402.

• 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, Springer-Verlag, New York (1995), 51-61.

• 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), 163-201.

• 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), 15-45.

• C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), 628-666.

• C. Heil and G. Kutyniok, Density of frames and Schauder bases of windowed exponentials, Houston J. Math., 34 (2008), 565-600.

• 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), 71-102.

• C. Heil and G. Kutyniok, The Homogeneous Approximation Property for wavelet frames, J. Approx. Theory, 147 (2007), 28-46.

• C. Heil, History and evolution of the Density Theorem for Gabor frames, J. Fourier Anal. Appl., 13 (2007), 113-166.

• R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames, Electron. Res. Announc. Amer. Math. Soc., 12 (2006), 71-86.

• 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), 307-344.   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), 105-143. 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. 479-493.

• R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Excesses of Gabor frames, Appl. Comput. Harmon. Anal., 14 (2003), 87-106.

• R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Deficits and excesses of frames, Adv. Comput. Math., Special Issue on Frames, 18 (2003), 93-116.

• 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), 153-164.

• O. Christensen, B. Deng, and C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., 7 (1999), 292-304.   Errata.

• Á. Bényi, K. Gröchenig, C. Heil, and K. Okoudjou, Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory, 54 (2005), 387-399.

• K. Gröchenig and C. Heil, Counterexamples for boundedness of pseudodifferential operators, Osaka J. Math., 41 (2004), 681-691.

• C. Heil, Integral operators, pseudodifferential operators, and Gabor frames, in: "Advances in Gabor Analysis," H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston (2003), 153-169.

• 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), 151-169.

• K. Gröchenig and C. Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, 34 (1999), 439-457.

• C. Heil, J. Ramanathan, and P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal., 150 (1997), 426-452.   Errata.

• R. Tinaztepe and C. Heil, Modulation spaces, BMO, and the Balian--Low Theorem, Sampl. Theory Signal Image Process., 11 (2012), 25--41.

• C. Heil and A. M. Powell, Regularity for complete and minimal Gabor systems on a lattice, Illinois J. Math., 53 (2010), 1077-1094.

• 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), 137-150.

• C. Heil and A. M. Powell, Gabor Schauder Bases and the Balian-Low Theorem, J. Math. Physics, 47 (2006).

• C. Heil, Linear independence of finite Gabor systems, in: "Harmonic Analysis and Applications," Birkhäuser, Boston (2006), 171-206.   Errata.

• K. Gröchenig, C. Heil, and K. Okoudjou, Gabor analysis in weighted amalgam spaces, Sampling Theory in Signal and Image Processing, 1 (2003), 225-259.

• K. Gröchenig, D. Han, C. Heil, G. Kutyniok, The Balian-Low theorem for symplectic lattices in higher dimensions, Appl. Comput. Harmon. Anal., 13 (2002), 169-176.

• K. Gröchenig and C. Heil, Gabor meets Littlewood-Paley: Gabor expansions in L^p(R^d), Studia Math., 146 (2001), 15-33.

• J. J. Benedetto, C. Heil, and D. F. Walnut, Gabor systems and the Balian-Low theorem, in: "Gabor Analysis and Algorithms: Theory and Applications," H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston (1998), 85-122.

• J. J. Benedetto, C. Heil, and D. F. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl., 1 (1995), 355-402.

• J. Benedetto, C. Heil, and D. Walnut, Uncertainty Principles for time-frequency operators, in: "Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations," Oper. Theory Adv. Appl. 58, I. Gohberg, ed., Birkhäuser, Basel (1992), 1-25.

• 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), 441-454.

• 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), 628-666.

• 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 unit-norm tight frames in Banach spaces, Graduate J. Math., 7 (2022), 17-38.

• G. J. Yoon and C. Heil, Duals of windowed exponential systems, Acta Appl. Math., 119 (2012), 97-112. The published version is available at www.springerlink.com. Here is a related talk from the Codex seminar series.

• A. Aldroubi, C. Cabrelli, C. Heil, K. Kornelson, and U. Molter, Invariance of a shift-invariant space, J. Fourier Anal. Appl., 16 (2012), 60-75.

• S. Bishop, C. Heil, Y. Y. Koo, and J. K. Lim, Invariances of frame sequences under perturbation, Linear Algebra Appl., 432 (2010), 1501--1514.

• C. Heil, Y. Y. Koo, and J. K. Lim, Duals of frame sequences, Acta Appl. Math., 107 (2009), 75-90. 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), 77-92.

• O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr., 186 (1997), 33-47.

• C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), 628-666.

• 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, Self-similarity 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), 23-39.

• C. Cabrelli, C. Heil, and U. Molter, Accuracy of several multi-dimensional refinable distributions, J. Fourier Anal. Appl., 6 (2000), 483-502.

• 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), 395-406.

• 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., Springer-Verlag, Singapore (1999), 121-161.

• C. Cabrelli, C. Heil, and U. Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory, 95 (1998), 5-52.   Errata.

• C. Heil and D. Colella, Matrix refinement equations: Existence and uniqueness, J. Fourier Anal. Appl., 2 (1996), 363-377.

• C. Heil, G. Strang and V. Strela, Approximation by translates of refinable functions, Numerische Math., 73 (1996), 75-94.   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, Springer-Verlag, New York (1995), 51-61.

• 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), 151-158.

• 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), 163-201.

• D. Colella and C. Heil, Characterizations of scaling functions: Continuous solutions, SIAM J. Matrix Anal. Appl., 15 (1994), 496-518.

• 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), 19-38.

• D. Colella and C. Heil, The characterization of continuous, four-coefficient scaling functions and wavelets, IEEE Trans. Inform. Theory, 38 (1992), 876-881.

• 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), 15-45.

• C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), 628-666.

• R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt, Pseudodifferential operators, microlocal analysis and image restoration, in: "Advances in Pseudo-Differential Operators," R. Ashino, P. Boggiatto, and M.-W. Wong, eds., Birkhäuser, Boston (2004), 187-202.

• 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), 1551-1579.

• R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt, Microlocal analysis, smooth frames and denoising in Fourier space, Asian Information-Science-Life, 1 (2002), 153-160.

• R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, and R. Vaillancourt, Microlocal filtering with multiwavelets, Comput. Math. Appl., 41 (2001), 111-133.

• 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), 548-563.

• C. Heil, The Wiener transform on the Besicovitch spaces, Proc. Amer. Math. Soc., 127 (1999), 2065-2071.

• 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), 74460K1--74460K8.

• 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), 209-217.

• 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), 39-46.

• 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), 101-118.

• 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), 9-20.

• 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), 293-302.

• 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), 663-667 (Section 7.15.5 in the 31st Edition, 2003, 723-726).

• C. Heil, Existence and accuracy for matrix refinement equations, Z. Angew. Math. Mech., Special issue on Applied Stochastics and Optimization, 76 (1996), 251-254.

• 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), 1796-1799.

• C. Heil, J. Ramanathan, and P. Topiwala, Asymptotic singular value decay of time-frequency 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), 15-24.

• C. Heil, Applications of the fast wavelet transform, in: "Advanced Signal-Processing Algorithms, Architectures, and Implementations" (San Diego, CA, 1990), Proc. SPIE Vol. 1348, F. T. Luk, ed., SPIE, Bellingham, WA (1990), 248-259.

• 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, Springer-Verlag, New York (1990), 147-160.

• C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), 628-666.

• Review of "Ten Lectures on Wavelets" by I. Daubechies (review appeared in SIAM Review, 35 (1993), 666-669).

• Review of "A First Course in Fourier Analysis," by D. W. Kammler (review appeared in SIAM Review, 43 (2001) 722-724).