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References
Ackiezer NI, Theory of Approximation, Ungar, New York, 1956. Ackiezer NI and Glazman IM, Theory of Linear Operators in Hilbert Space, Ungar,
New York, 1961. Aslaksen EW and Klauder JR, Unitary representations of the affine group, J. Math.
Phys. 9(1968), 206-211. Aslaksen EW and Klauder JR, Continuous representation theory using the affine group,
J. Math. Phys. 10(1969), 2267-2275. Auslander L and Gertner I, Wideband ambiguity functions and the a • x + b group, in
Auslander L, Griinbaum F A, Helton J W, Kailath T, Khargonekar P, and Mitter S, eds., Signal Processing: Part I - Signal Processing Theory, Springer-Verlag, New York, pp. 1-12, 1990.
Backus J, The Acoustical Foundations of Music, second edition, Norton, New York, 1977.
Bacry H, Grossmann A, and Zak J, Proof of the completeness of lattice states in the kq-representation, Phys. Rev. B 12(1975), 1118-1120.
Balian R, Un principe d'incertitude fort en theorie du signal ou en mecanique quan-tique, C. R. Acad. Sci. Paris, 292(1981), Serie 2.
Baraniuk RG and Jones DL, New orthonormal bases and frames using chirp functions, IEEE Transactions on Signal Processing 41(1993), 3543-3548.
Bargmann V and Wigner EP, Group theoretical discussion of relativistic wave equations, Proc. Nat. Acad. Sci. USA 34(1948), 211-233.
Bargmann V, Butera P, Girardello L, and Klauder JR, On the completeness of coherent states, Reps. Math. Phys. 2(1971), 221-228.
Barton DK, Modern Radar System Analysis, Artech House, Norwood, MA, USA, 1988. Bastiaans MJ, Gabor's signal expansion and degrees of freedom of a signal, Proc. IEEE
68(1980), 538-539. Bastiaans MJ, A sampling theorem for the complex spectrogram and Gabor's expan
sion of a signal in Gaussian elementary signals, Optical Engrg. 20(1981), 594-598. Bastiaans MJ, Gabor's signal expansion and its relation to sampling of the sliding-
window spectrum, in Marks II JR (1993), ed., Advanced Topics in Shannon Sampling and Interpolation Theory, Springer-Verlag, Berlin, 1993.
Bateman H, The transformation of the electrodynamical equations, Proc. London Math. Soc. 8(1910), 223-264.
Battle G, A block spin construction of ondelettes. Part I: Lemarie functions, Comm. Math. Phys. 110(1987), 601-615.
Battle G, Heisenberg proof of the Balian-Low theorem, Lett. Math. Phys. 15(1988), 175-177.
Battle G, Wavelets: A renormalization group point of view, in Ruskai MB, Beylkin G, Coifman R, Daubechies I, Mallat S, Meyer Y, and Raphael L, eds., Wavelets and their Applications, Jones and Bartlett, Boston, 1992.
Benedetto JJ and Walnut DF, Gabor frames for L2 and related spaces, in Benedetto JJ and Frazier MW, eds., Wavelets: Mathematics and Applications, CRC Press, Boca Raton, 1993.
G. Kaiser, A Friendly Guide to Wavelets, Modern Birkhäuser Classics, 291 DOI 10.1007/978-0-8176-8111-1, © Gerald Kaiser 2011
292 A Friendly Guide to Wavelets
Benedetto JJ and Frazier MW, eds., Wavelets: Mathematics and Applications, CRC Press, Boca Raton, 1993.
Bernfeld M, CHIRP Doppler radar, Proc. IEEE 72(1984), 540-541. Bernfeld M, On the alternatives for imaging rotational targets, in Radar and Sonar,
Part II, Griinbaum FA, Bernfeld M and Bluhat RE, eds., Springer-Verlag, New York, 1992.
Beylkin G, Coifman R, and Rokhlin V, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math. 44(1991), 141-183.
Bialynicki-Birula I and Mycielski J, Uncertainty relations for information entropy, Commun. Math. Phys. 44(1975), 129.
de Boor C, DeVore RA, and Ron A, Approximation from shift-invariant subspaces of L 2 (R d ) , Trans. Amer. Math. Soc. 341(1994), 787-806.
Born M and Wolf E, Principles of Optics, Pergamon, Oxford, 1975. Chui CK, An Introduction to Wavelets, Academic Press, New York, 1992a. Chui CK, ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, New
York, 1992b. Chui CK and Shi X, Inequalities of Littlewood-Paley type for frames and wavelets,
SIAM J. Math. Anal. 24(1993), 263-277. Cohen A, Ondelettes, analyses multiresolutions et filtres miroir en quadrature, Inst.
H. Poincare, Anal, non lineare 7(1990), 439-459. Coifman R, Meyer Y, and Wickerhauser MV, Wavelet analysis and signal processing, in
Ruskai MB, Beylkin G, Coifman R, Daubechies I, Mallat S, Meyer Y, and Raphael L, eds., Wavelets and their Applications, Jones and Bartlett, Boston, 1992.
Coifman R and Rochberg R, Representation theorems for holomorphic and harmonic functions in Lp, Asterisque 77(1980), 11-66.
Cook CE and Bernfeld M, Radar Signals, Academic Press, New York; republished by Artech House, Norwood, MA, 1993 (1967).
Cunningham E, The principle of relativity in electrodynamics and an extension thereof, Proc. London Math. Soc. 8(1910), 77-98.
Daubechies I, Grossmann A, and Meyer Y, Painless non-orthogonal expansions, J. Math. Phys. 27(1986), 1271-1283.
Daubechies I, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41(1988), 909-996.
Daubechies I, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36(1990), 961-1005.
Daubechies I, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. Deslauriers G and Dubuc S, Interpolation dyadique, in Fractals, dimensions non
enti'eres et applications, G. Cherbit, ed., Masson, Paris, pp. 44-45, 1987. Dirac PA M, Quantum Mechanics, Oxford University Press, Oxford, 1930. Duffin RJ and Schaeffer AC, A class of nonharmonic Fourier series, Trans. Amer. Math.
Soc. 72(1952), 341-366. Einstein A, Lorentz HA, Weyl H, The Principle of Relativity, Dover, 1923. Feichtinger HG and Grochenig K, A unified approach to atomic characterizations via
integrable group representations, in Proc. Conf. Lund, June 1986, Lecture Notes in Math. 1302, (1986).
References 293
Feichtinger HG and Grochenig K, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86(1989), 307-340, 1989a.
Feichtinger HG and Grochenig K, Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsch. f. Mathematik 108(1989), 129-148, 1989b.
Feichtinger HG and Grochenig K, Gabor wavelets and the Heisenberg group: Gabor expansions and short-time Fourier transform from the group theoretical point of view, in Chui CK (1992), ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, New York, 1992.
Feichtinger HG and Grochenig K, Theory and practice of irregular sampling, in Benedetto JJ and Frazier MW (1993), eds., Wavelets: Mathematics and Applications, CRC Press, Boca Raton, 1993.
Folland GB, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ, 1989.
Gabor D, Theory of communication, J. Inst. Electr. Eng. 93(1946) (III), 429-457. Gel'fand IM and Shilov GE, Generalized Functions, Vol. I, Academic Press, New York,
1964. Glimm J and Jaffe A (1981) Quantum Physics: A Functional Point of View, Springer-
Verlag, New York. Gnedenko BV and Kolmogorov AN, Limit Distributions for Sums of Independent Ran
dom Variables, Addison-Wesley, Reading, MA, 1954. Grochenig K, Describing functions: atomic decompositions versus frames, Monatsch.
f Math. 112(1991), 1-41. Grochenig K, Acceleration of the frame algorithm, to appear in IEEE Trans. Signal
Proc, 1993. Gross L, Norm invariance of mass-zero equations under the conformal group, J. Math.
Phys. 5(1964), 687-695. Grossmann A and Morlet J, Decomposition of Hardy functions into square-integrable
wavelets of constant shape, SI AM J. Math. Anal. 15(1984), 723-736. Heil C and Walnut D, Continuous and discrete wavelet transforms, SI AM Rev. 31
(1989), 628-666. Hill EL, On accelerated coordinate systems in classical and relativistic mechanics,
Phys. Rev. 67(1945), 358-363. Hill EL, On the kinematics of uniformly accelerated motions and classical electromag
netic theory, Phys. Rev. 72(1947), 143-149; Hill EL, The definition of moving coordinate systems in relativistic theories, Phys.
Rev. 84(1951), 1165-1168. Hille E, Reproducing kernels in analysis and probability, Rocky Mountain J. Math.
2(1972), 319-368. Jackson JD, Classical Electrodynamics, Wiley, New York, 1975. Jacobsen HP and Vergne M, Wave and Dirac operators, and representations of the
conformal group, J. Fund. Anal. 24(1977), 52-106. Janssen AJEM, Gabor representations of generalized functions, J. Math. Anal. Appl.
83(1981), 377-394.
294 A Friendly Guide to Wavelets
Kaiser G, Phase-Space Approach to Relativistic Quantum, Mechanics, Thesis, University of Toronto Department of Mathematics, 1977a.
Kaiser G, Phase-space approach to relativistic quantum mechanics, Part I: Coherent-state representations of the Poincare group, J. Math. Phys. 18(1977), 952-959, 1977b.
Kaiser G, Phase-space approach to relativistic quantum mechanics, Part II: Geometrical aspects, J. Math. Phys. 19(1978), 502-507, 1978a.
Kaiser G, Local Fourier analysis and synthesis, University of Lowell preprint (unpublished). Originally NSF Proposal #MCS-7822673, 1978b.
Kaiser G, Phase-space approach to relativistic quantum mechanics, Part III: Quantization, relativity, localization and gauge freedom, J. Math. Phys. 22(1981), 705-714.
Kaiser G, A sampling theorem in the joint time-frequency domain, University of Lowell preprint (unpublished), 1984.
Kaiser G, Quantized fields in complex spacetime, Ann. Phys. 173(1987), 338-354. Kaiser G, Quantum Physics, Relativity, and Complex Spacetime: Towards a New Syn
thesis, North-Holland, Amsterdam, 1990a. Kaiser G, Generalized wavelet transforms, Part I: The windowed X-ray transform,
Technical Reports Series #18, Mathematics Department, University of Lowell. Part II: The multivariate analytic-signal transform, Technical Reports Series #19, Mathematics Department, University of Lowell, 1990b.
Kaiser G, An algebraic theory of wavelets, Part I: Complex structure and operational calculus, SIAM J. Math. Anal. 23(1992), 222-243, 1992a.
Kaiser G, Wavelet electrodynamics, Physics Letters A 168(1992), 28-34, 1992b. Kaiser G, Space-time-scale analysis of electromagnetic waves, in Proc. of IEEE-SP
Internat. Symp. on Time-Frequency and Time-Scale Analysis, Victoria, Canada, 1992c.
Kaiser G and Streater RF, Windowed Radon transforms, analytic signals and the wave equation, in Chui CK, ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, New York, pp. 399-441, 1992.
Kaiser G, Wavelet electrodynamics, in Meyer Y and Roques S, eds., Progress in Wavelet Analysis and Applications, Editions Frontieres, Paris, pp. 729-734 (1993).
Kaiser G, Deformations of Gabor frames, J. Math. Phys. 35(1994), 1372-1376, 1994a. Kaiser G, Wavelet Electrodynamics: A Short Course, Lecture notes for course given
at the Tenth Annual ACES (Applied Computational Electromagnetics Society) Conference, March 1994, Monterey, CA, 1994b.
Kaiser G, Wavelet electrodynamics, Part II: Atomic composition of electromagnetic waves, Applied and Computational Harmonic Analysis 1(1994), 246-260 (1994c).
Kaiser G, Cumulants: New Path to Wavelets? UMass Lowell preprint, work in progress, (1994d).
Kalnins EG and Miller W, A note on group contractions and radar ambiguity functions, in Grunbaum FA, Bernfeld M, and Bluhat RE, eds., Radar and Sonar, Part II, Springer-Verlag, New York, 1992.
Katznelson Y, An Introduction to Harmonic Analysis, Dover, New York, 1976. Klauder JR and Surarshan ECG, Fundamentals of Quantum Optics, Benjamin, New
York, 1968.
References 295
Lawton W, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32(1991), 57-61.
Lemarie PG, Une nouvelle base d'ondelettes de L 2 (R n ) , J. de Math. Pure et Appl. 67(1988), 227-236.
Low F, Complete sets of wave packets, in A Passion for Physics - Essays in Honor of Godfrey Chew, World Scientific, Singapore, pp. 17-22, 1985.
Lyubarskii Yu I, Frames in the Bargmann space of entire functions, in Entire and Subharmonic Functions, Advances in Soviet Mathematics 11(1989), 167-180.
Maas P, Wideband approximation and wavelet transform, in Griinbaum FA, Bernfeld M, and Bluhat RE, eds., Radar and Sonar, Part II, Springer-Verlag, New York, 1992.
Mallat S, Multiresolution approximation and wavelets, Trans. Amer. Math. Soc. 315 (1989), 69-88.
Mallat S and Zhong S, Wavelet transform scale maxima and multiscale edges, in Ruskai MB, Beylkin G, Coifman R, Daubechies I, Mallat S, Meyer Y, and Raphael L, eds., Wavelets and their Applications, Jones and Bartlett, Boston, 1992.
Messiah A, Quantum Mechanics, North-Holland, Amsterdam, 1961. Meyer Y, Wavelets and Operators, Cambridge University Press, Cambridge, 1993a. Meyer Y, Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993b. Meyer Y and Roques S, eds., Progress in Wavelet Analysis and Applications, Editions
Frontieres, Paris, 1993. Miller W, Topics in harmonic analysis with applications to radar and sonar, in Bluhat
RE, Miller W, and Wilcox CH, eds., Radar and Sonar, Part I, Springer-Verlag, New York, 1991.
Morlet J, Sampling theory and wave propagation, in NATO ASI Series, Vol. I, Issues in Acoustic Signal/Image Processing and Recognition, Chen CH, ed., Springer-Verlag, Berlin, 1983.
Moses HE, Eigenfunctions of the curl operator, rotationally invariant Helmholtz theorem, and applications to electromagnetic theory and fluid mechanics, SIAM J. Appl. Math. 21(1971), 114-144.
Naparst H, Radar signal choice and processing for a dense target environment, Ph. D. thesis, University of California, Berkeley, 1988.
Page L, A new relativity, Phys. Rev. 49(1936), 254-268. Papoulis A, The Fourier Integral and its Applications, McGraw-Hill, New York, 1962. Papoulis A, Signal Analysis, McGraw-Hill, New York, 1977. Papoulis A, Probability, Random Variables, and Stochastic Processes, McGraw-Hill,
New York, 1984. Rihaczek AW, Principles of High-Resolution Radar, McGraw-Hill, New York, 1968. Roederer JG (1975), Inroduction to the Physics and Psychophysics of Music, Springer-
Verlag, Berlin. Rudin W, Real and Complex Analysis, McGraw-Hill, New York, 1966. Riihl W, Distributions on Minkowski space and their connection with analytic repre
sentations of the conformal group, Commun. Math. Phys. 27(1972), 53-86. Ruskai MB, Beylkin G, Coifman R, Daubechies I, Mallat S, Meyer Y, and Raphael L,
eds., Wavelets and their Applications, Jones and Bartlett, Boston, 1992.
296 A Friendly Guide to Wavelets
Schatzberg A and Deveney AJ, Monostatic radar equation for backscattering from dynamic objects, preprint, AJ Deveney and Associates, 355 Boylston St., Boston, MA 02116; 1994.
Seip K and Wallsten R, Sampling and interpolation in the Bargmann-Fock space, preprint, Mittag-Leffler Institute, 1990.
Shannon CE, Communication in the presence of noise, Proc. IRE, January issue, 1949. Smith MJ T and Banrwell TP, Exact reconstruction techniques for tree-structured
subband coders, IEEE Trans. Acoust. Signal Speech Process. 34(1986), 434-441. Stein E, Singular Integrals and Differentiability Properties of Functions, Princeton
University Press, Princeton, NJ, 1970. Stein E and Weiss G (1971), Fourier Analysis on Euclidean Spaces, Princeton Univer
sity Press, Princeton, NJ. Strang G and Fix G, A Fourier analysis of the finite-element method, in Constructive
Aspects of Functional Analysis, G. Geymonat, ed., C.I.M.E. II, Ciclo 1971, pp. 793-840, 1973.
Strang G, Wavelets and dilation equations, SIAM Review 31(1989), 614-627. Strang G, The optimal coefficients in Daubechies wavelets, Physica D 60(1992), 239-
244. Streater RF and Wightman AS, PCT, Spin and Statistics, and All That, Benjamin,
New York, 1964. Strichartz RS, Construction of orthonormal wavelets, in Benedetto JJ and Frazier MW,
eds., Wavelets: Mathematics and Applications, CRC Press, Boca Raton, 1993. Swick DA, An ambiguity function independent of assumption about bandwidth and
carrier frequency, NRL Report 6471, Washington, DC, 1966. Swick DA, A review of wide-band ambiguity functions, NRL Report 6994, Washington,
DC, 1969. Tewfik AH and Hosur S, Recent progress in the application of wavelets in surveillance
systems, Proc. SPIE Conf. on Wavelet Applications, 1994. Vetterli M, Filter banks allowing perfect reconstruction, Signal Process. 10(1986), 219-
244. Ward RS and Wells RO, Twistor Geometry and Field Theory, Cambridge University
Press, Cambridge, 1990. Woodward PM, Probability and Information Theory, with Applications to Radar, Perg-
amon Press, London, 1953. Young RM, An Introduction to Nonharmonic Fourier Series, Academic Press, New
York, 1980. Zakai M, Information and Control 3(1960), 101. Zou H, Lu J, and Greenleaf F, Obtaining limited diffraction beams with the wavelet
transform, Proc. IEEE Ultrasonic Symposium, Baltimore, MD, 1993a. Zou H, Lu J, and Greenleaf F, A limited diffraction beam obtained by wavelet theory,
preprint, Mayo Clinic and Foundation 1993b.
Index
The keywords cite the main references only.
accelerating reference frame, 212, 244 acoustic wavelets, 273
absorbed, 278 emitted, 278
adjoint, 13, 24-25 admissibility condition, 67 admissible, 264 affine
group, 205, 257 transformations, 257
aliasing operator, 162 almost everywhere (a.e.), 22 ambiguity function, 250, 254
conformal cross, 266 self, 266
narrow-band, 254 cross, 254 self, 254
wide-band, 248 cross, 250 self, 250
analytic signal, 68, 206, 218, 219 lower 73 upper 73
analyzing operator, 83 antilinear, 12 atomic composition, 233 averaging
operator, 142 property, 141
Balian-Low theorem, 116 band-limited, 99 bandwidth, 99, 252 basis, 4
dual, 9 reciprocal, 16
Bernstein's inequality, 101
biorthogonal, 15 blurring, 144, 242 bounded, 26
linear functional, 27 bra, 14
carrier frequency, 252 causal, 205
cone, 222 tube, 206, 223
characteristic function, 40 chirp, 47 complete, 4, 80 complex
space-time, 206 structure, 207
conformal cross-ambiguity function, 250, 254,
266 group, 204 self-ambiguity function, 266 transformations, 260
special, 260, 269 consistency condition, 57, 87, 234, 249 counting measure, 40, 106 cross-ambiguity function, 250, 254 cumulants, 191-196
de-aliasing operator, 163, 164 dedicated wavelets, 204 demodulation, 253 detail, 63 diadic
interpolation, 189 rationals, 188, 190
differencing operator, 152 dilation equation, 142 Dirac notation, 14 distribution, 26
298 Index
Doppler effect, 243, 247 shift, 252
down-sampling operator, 149 duality relation, 9 duration, 206, 242
electromagnetic wavelets, 224 elementary
current, 279 scatterers, 263 scattering event, 265
Euclidean region, 225 evaluation map, 224 exact scaling functions, 191
filter, coefficients, 142 extremal-phase, 181 linear-phase, 181 maximum-phase, 181 minimum-phase, 181
finite impulse response (FIR), 155, focal point, 205 Fourier
coefficients, 27 series, 26 transform, 29
fractal, 183 frame, 82
bounds, 82 condition, 82 deformations of, 117-218 discrete 82, 91 generalized, 82 normalized 83 operator, 83 reciprocal, 86 snug 90 tight 82 vectors, 82
future cone, 222
Gabor expansions, 114 gauge freedom, 212
Haar wavelet, 159 harmonic oscillator, 118 helicity, 206, 210
high-pass filter, 147 sequence, 154
Hilbert space, 23 transform, 251, 252
Holder continuous, 187-188 exponents, 188
hyperbolic lattice, 124
ideal band-pass filter, 167 ideal low-pass filter, 110 identity operator, 10 image, 6 incomplete, 4 inner product, 12
standard, 12 instantaneous frequency, 47 interpolation operator, 148 inverse
Fourier transform, 33 image, 36 problem, 248
isometry, 69 partial 69
ket, 14 Kronecker delta, 5
least-energy representation, 101 least-squares approximation, 88 Lebesgue
integral, 21 measure, 35
light cone, 208 linear, 6
functional, 24 bounded 24
linearly dependent, 4, 80 independent, 4
Lipschitz continuous, 187 Lorentz
condition, 211 group, 259 invariant inner product, 208 transformations, 237
low-pass filter, 147
Index 299
massless Sobolev space, 216 matching, 248 Maxwell's equations, 207 mean time, 191 measurable, 21, 35 measure, 21, 35
space, 40 zero, 22
metric operator, 18 Mexican hat, 75 Meyer wavelets, 167 moments, 193 multiresolution analysis, 144
narrow-band signal, 252 negative-frequency cone, 222 nondiffracting beams, 272 norm, 12 numerically stable, 104 Nyquist rate, 101
operator, 23 bounded, 23
orthogonal, 13 orthonormal, 13 oscillation, 273 over complete, 4 oversampling, 88, 101
Parseval's identity, 33 past cone, 222 periodic, 26 Plancherel's theorem, 29, 32 plane waves, 206 Poincare group, 259 polarization identity, 33 positive-definite operators, 18 potentials, 211 proxies, 246, 256, 280
quadrature mirror filters, 148 quaternionic structure, 171
range, 245 ray filter, 243 redundancy, 108 reference wavelet, 239, 240n., 241 reflectivity distribution, 248
reflector complex, 267 elementary, 263 standard, 263
regular lattice, 117 representation, 206, 261 reproducing kernel, 70, 228-233 resolution, 145 resolution of unity, 10
continuous, 55 rest frame, 281-283 Riesz representation theorem, 24
sampling, 92, 99 rate, 101
sampling operator, 161 scale factor, 63 scaling
function, 140 invariant, 126 periodic, 127
Schwarz inequality, 18, 23 Shannon
sampling theorem, 99 wavelets, 167
space-time domain, 31 inversion, 260
span, 9 special conformal transformations, 212 spectral factorization, 180 square-integrable, 23 square-summable, 20 stability, 243
condition, 128 star notation, 17 subband, 111
filtering, 153-155 subspace, 4 support, 23
compact, 23 synthesizing operator, 85, 236
time-limited, 102 transform,
analytic-signal, 218 triangle inequality, 18, 23 trigonometric polynomial, 143
300 Index
tube, causal, 206, 223 future, 223 past, 223
uncertainty principle, 52 up-sampling operator, 148
vector space, 4 complex 4 real 4
wave equation, 207, 272 vector, 30
wavelet, 62 basic 62 mother 62 packets, 172 space, 236
wave number-frequency domain, 31 Weyl-Heisenberg group, 51, 205 window, 45
Zak transform, 117 zero moments, 178 zoom operators, 148