Modern tools for weyl-heisenberg (gabor) frame theory

Advances in Imaging and Electron Physics

Volumn 115, Issue C, 2001, Pages 1-127

  Casazza, Peter G ^a  

^a UNIVERSITY OF MISSOURI   (United States)

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[No Author keywords available]

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EID: 0002362830     PISSN: 10765670     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1076-5670(01)80094-X     Document Type: Article

References (177)

1. (No. 6). Aldroubi A. Portraits of frames. Proceedings AMS 123 (1995) 1661-1668
2. Aldroubi A. Oblique projections in atomic spaces. Proc. Amer. Math. Soc. 124 (1996) 2051-2060
3. Aldroubi, A. and Gröchenig, K. (To appear). Beurling-Landu-type theorems for non-uniform sampling in shift invariant spline spaces. Jour. Fourier Anal. and Appl.s
4. Aldroubi A., and Unser M. Sampling procedures in function spaces and asymptotic equivalent with Shannon's sampling theory. Numer. Funct. Anal. and Optimiz. 15 1&2 (1994) 1-21
5. p.
6. Anderson J.H. Commutators in ideals of trace-class operators, II. Indiana Univ. Math. J. 35 (1986) 373-378
7. Auslander L., and Tolimieri R. Is computing with the finite Fourier transform pure or applied mathematics. Bull. AMS 1 (1974) 847-897
8. Auslander L., and Tolimieri R. Radar ambiguity functions and group theory. SIAM J. Math. Anal. 16 (1985) 577-599
9. Bacry H., Grossmann A., and Zak J. Proof of completeness of lattice states in the kq representation. Phys. Rev. B 12 (1975) 1118-1120
10. Balan R. Stability theorems for Fourier frames and wavelet Riesz bases. Jour. Fourier Anal. and Appls. 3 5 (1997) 499-504
11. Balan, R. and Landan, Z. private communication.
12. Balan, R. and Landau, Z. private communication 2.
13. Balian R. Un principe d'incertitude fort en théorie du signal on en mécanique quantique. C.R. Acad. Sci. Paris 292 (1981) 1357-1362
14. Bastiaans M.J. Gabor's expansion of a signal into Gaussian elementary signals. Proceedings of the IEEE 68 (1980) 583
15. Bastianns M.J. A sampling theorem for the complex spectrogram and Gabor's expansions of a signal in Gaussian elementary signals. Opt. Eng. 20 4 (1981) 594-598
16. Battle G. Heisenberg proof of the Balian-Low theorem. Lett. Math. Phys. 15 (1988) 175-177
17. Benedetto J. Gabor representation and wavelets. In: Colella D. (Ed). Commutative Harmonic Analysis. Contemp. Math. 9 (1989), AMS, Providence 9-27
18. Benedetto, J., Frame decompositions, sampling, and uncertainty principle inequalities, preprint.
19. In: Benedetto J., and Frazier M. (Eds). Wavelets: Mathematics and Applications, in Studies in Advanced Mathematics (1994), CRC Press, Boca Ratton, FL
20. Benedetto J., Heil C., and Walnut D. Differentiation, and the Ballian-Low Theorem. Jour. Fourier Anal. and Appls 1 No. 4 (1995) 355-402
21. Benedetto J., and Li S. The theory of multiresolution analysis frames and applications to filter banks. Appl. Comp. Har. Anal. 5 4 (1998) 389-427
22. Benedetto J., and Treiber O.M. A perfect reconstruction multirate system with narrow band filters. In: Debnath L. (Ed). Wavelet Analysis and its Applications (1999), Birkhauser, Boston
23. Benedetto, J. and Treiber, O. M. Wavelet frames: multiresolution analysis and extension principles, to appear, Wavelet transforms and time-frequency signal analysis, L Debnath, ed.
24. Benedetto, J. and Scott, S. Frames, irregular sampling, and a wavelet auditory model, to appear: Sampling Theory and Practice, F. Marvasti, ed. Plenum.
25. 2 and related spaces. In: Benedetto J.J., and Frazier M.W. (Eds). Wavelets Mathematics and Applications (1994), CRC Press Inc., Boca Raton, FL
26. Berberian S.K. Introduction to Hilbert Space (1961), Oxford Univ. Press, Oxford
27. Bergmans J.W.M., and Janssen A.J.E.M. Robust data equalization, fractional tap spacing and the Zak transform. Phillips J. Res. 42 (1987) 351-398
28. Boon M., and Zak J.H. Amplitudes on von Neumann lattices. J. Math. Phys. 22 (1981) 1090-1099
29. Boon M., Zak J., and Zucker I.J. Rational von Neumann lattices. J. Math. Phys. 24 (1983) 316-323
30. d. Trans. Amer. Math. Soc. 341 (1994) 787-806
31. de Boor C., DeVore R., Ron A., and Shen Z. The Structure of shift invariant spaces and applications to approximation theory. J. Functional Anal. No. 119 (1994) 37-78
32. Borwein, P. and Erdelyi, T. On the zeroes of polynomials with restricted coefficients, preprint.
33. Borwein, P., Erdelyi, T. and Kos, G. Littlewood-type problems on [0,1], preprint.
34. 2(ℝ), preprint
35. Casazza P.G. Characterizing Hilbert space frames with the subframe property. Illinois. Jour. Math. 41 4 (1997) 648-666
36. Casazza P.G. Every frame is a sum of three (but not two) orthonormal bases and other frame representations. Jour. Fourier Anal. and Appls. 4 No. 6 (1998) 727-732
37. Casazza, P. G. The Art of frame theory, to appear, Taiwan Jour. of Math.
38. Casazza P.G. Local theory of frames and Schauder bases for Hilbert space. Illnois Jour. Math. 43 No. 2 (1999) 291-306
39. 0. Jour. of Math. Anal. and Appls. 202 (1996) 940-950
40. Casazza P.G., and Christensen O. Approximation of the frame coefficients using finite dimensional methods. Jour. of Electronic Imaging, special Volume "Best papers in electronic imaging 1997" 6 4 (1997) 479-483
41. Casazza P.G., and Christensen O. Perturbation of operators and applications to frame theory. Jour. Fourier Anal. and Appls. 3 5 (1997) 543-557
42. Casazza P.G., and Christensen O. Frames containing a Riesz basis and preservation of this property under perturbations. SIAM Jour. Math. Anal. 29 1 (1998) 266-278
43. Casazza P.G., and Christensen O. Riesz frames and approximation of the frame coefficients. Jour. Approx. Theory and Appls 14 2 (1998) 1-11
44. Casazza P.G., and Christensen O. Frames and Schauder bases, Approximation Theory. In: Mohapatra R.N., Nashed Z., Sharma A., and Szabados J. (Eds). Memory of A. K. Varna, N. K. Govil (1998), Marcel Dekker 133-139
45. 2(ℝ). To appear, Proc. AMS.
46. Casazza P.G., and Christensen O. Approximation of the inverse frame operator. In: Chui C., and Schumaker L. (Eds). Approximation Theory IV, Vol. 1: Theoretical Aspects (1999), Vanderbilt University Press 67-74
47. Casazza P.G., Christensen O., and Janssen A.J.E.M. Classifying tight Weyl-Heisenberg frames. In: Baggett L.W., and Larson D.R. (Eds). The Functional and Harmonic Analysis of Wavelets and Frames. Contemp. Math. 247 (2000) 131-148
48. Casazza, P. G., Christensen. O. and Janssen, A. J. E. M. Weyl-Heisenberg frames, translation-invariant systems, and the Walnut representation, preprint.
49. Casazza, P. G., Christensen, O. and Kalton, N. J. Frames of translates, preprint.
50. Casazza, P. G., Christensen. O. and Lammers, M. Perturbation theory for Weyl-Heisenberg frames, preprint.
51. Casazza P.G., Han D., and Larson D.R. Frames for Banach Spaces. In: Baggett L.W., and Larson D.R. (Eds). The Functional and Harmonic Analysis of Wavelets and Frames. Contemp. Math. 247 (2000) 149-182
52. Casazza P.G., and Kalton N.J. Generalizing the Paley-Wiener perturbation theory for Banach spaces. Proc. AMS 127 2 (1999) 519-527
53. Casazza, P. G. and Kalton, N. J. A classical problem in complex function theory is equivalent to classifying Weyl-Heisenberg frame sets, preprint.
54. Casazza, P. G. and Lammers, M. C. The pointwise inner product and Weyl-Heisenberg frames, preprint.
55. Casazza, P. G. and Lammers, M. C. Analyzing the Weyl-Heisenberg frame identity, preprint.
56. Christensen O. Frames and the projection method. Appl. Comp. Harmonic Anal. 1 (1993) 50-53
57. Christensen O. A Paley-Wiener theorem for frames. Proc. AMS 123 (1995) 2199-2202
58. Christensen O. Frame perturbations. Proc. Amer. Math. Soc. 123 (1995) 1217-1220
59. Christensen O. Frames and pseudo-inverses. J. Math. Anal. Appl. 195 (1995) 401-414
60. Christensen O. Frames containing a Riesz basis and approximation of the frame coefficients using finite dimensional methods. Jour. Math. Anal. and Appls. 199 (1996) 256-270
61. Christensen O. Atomic decomposition via projective group representations. Rocky Mountain Jour. Math. 26 4 (1996) 1289-1312
62. Christensen O. Moment problems and stability results for frames with applications to irregular sampling and Gabor frames. Appl. and Comp. Har. Anal. 3 (1996) 82-86
63. Christensen O. Perturbation of frames and applications to Gabor frames. In: Feichtingter H.G., and Strohmer T. (Eds). Gabor analysis and algorithms: Theory and applications (1998), Birkhauser, Boston 193-209
64. Christensen, O., Deng, B. and Heil, C. Density of Gabor frames, preprint.
65. Christensen O., and Heil C. Perturbations of Banach frames and atomic decompositions. Math. Nachr. 185 (1997) 33-47
66. Clasen T.A.C.M., and Mecklenbrauker W.F.G. Wigner distribution-A tool for time-frequency signal analysis. Phillips J. Res. 35 (1980) 217-250
67. Conway J.B. A Course in Functional Analysis, Graduate Texts in Math (1985), Springer-Verlag, New York
68. Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 409-996
69. Daubechies I. Time-frequency localization operators: A geometric phase space approach. IEEE Trans. Theory 34 (1988) 805
70. Daubechies I. The wavelet transition, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 36 5 (1990) 961-1005
71. Daubechies I. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Math. Philadelphia (1992)
72. Daubechies I. Gabor time-frequency lattices and the Wexler-Raz identity. Jour. Fourier Anal. and Appl. I 4 (1995) 437-478
73. Daubechies I., Grossmann A., and Meyer Y. Painless nonorthogonal expansions. J. Math. Phys. 27 (1986) 1271-1283
74. Daubechies I., Jaffard S., and Journé J.L. A simple Wilson orthonormal basis with exponential decay. SIAM J. Math. Anal. 22 No. 2 (1991) 554-572
75. Daubechies I., and Janssen A.J.E.M. Two theorems on lattice expansions. IEEE Trans. Inform. Theory 39 1 (1993) 3-6
76. Daubechies I., Landau H., and Landau Z. Gabor time-frequency lattices and the Wexler-Rax identity. J. Fourier Anal. and Appls. 1 No. 4 (1995) 437-478
77. Daubechies I., and Paul T. Time-frequency localization operators-A geometric phase space approach: II: The use of dilations. inverse problems 4 (1988) 661-680
78. DeVore R.A., Jawerth B., and Lucier B.J. Image compression through wavelet transform coding. IEEE Trans. on Inform. Theory 38 (1992) 719-746
79. Diestel J. Sequences and Series in Banach Spaces, Graduate Texts in Mathematics (1984), Springer-Verlag, New York
80. Ding J., and Huang L.J. On the perturbation of the least squares solutions in Hilbert spaces. Linear Algebra Appl. 212-213 (1994) 487-500
81. Ding J., and Huang L.J. Perturbation of generalized inverses of linear operator in Hilbert spaces. J. Math. Anal. Appl. 198 (1996) 505-516
82. Duffin R.J., and Schaeffer A.C. A class of nonharmonic Fourier series. Trans. AMS 72 (1952) 341-366
83. p spaces. Academic Press, Pure and Applied Math. Series 38 (1970)
84. Edwards R.E. Fourier Series: A Modern Introduction Vol. I (1979), Springer-Verlag, New York
85. Favier S.J., and Zalik R.A. On the stability of frames and Riesz bases. Appl. Comput. Harm. Anal. 2 (1995) 160-173
86. Feichtinger H. Banach convolution algebras of Wiener type, Functions, Series, Operators. In: Bolyai J. (Ed). Proc. Conf. Budapest 38 Colloq. Math. Soc. (1980) 509-524
87. Feichtinger H. Atomic characterizations of modulation spaces through Gabor-type representations. Constructive Function Theory-(Edmonton, 1986). Rocky Mountain Jour. Math. 19 1 (1989) 113-125
88. Feichtinger H., Christensen O., and Strohmer T. Group theoretical approach to Gabor analysis. Optical Engineering 34 6 (1995) 1697-1704
89. Feichtinger H., and Grobner P. Banach spaces of distributions defined by decomposition methods. I. Math. Nachr. 123 (1985) 97-120
90. Feichtinger H., and Gröchenig K. A unified approach to atomic decompositions through integrable group representations. In: Cwikel M., et al. (Ed). Function Spaces and Applications. Lecture Notes in Math 1302 (1988), Springer-Verlag, Berlin, N.Y.
91. Feichtinger H., and Gröchenig K. Banach spaces related to integrable group representations and their atomic decompositions. I. Monatsch. Math. 108 (1989) 129-148
92. Feichtinger H.G., and Gröchenig K. Gabor frames and time-Frequency analysis of distributions. Jour. of Functional Analysis 146 (1997) 464-495
93. Feichtinger, H. G. and Janssen, A. J. E. M. Validity of WH-frame bound conditions depends on lattice parameters, Appl. Conput. Harm. Anal., to appear.
94. In: Feichtinger H., and Strohmer T. (Eds). Gabor Analysis and Algorithms: Theory and Applications (1999), Birkhauser, Boston
95. Feichtinger H.G., and Zimmermann G. A Banach space of test functions for Gabor analysis. In: Feichtinger G., and Strohmer T. (Eds). Gabor Analysis and Algorithms: Theory and Applications (1999), Birkhauser, Boston 123-170
96. *-algebras, preprint.
97. Frank, M., Paulsen, V. I. and Tiballi, T. R. Symmetric approximation of frames and bases in Hilbert spaces, preprint.
98. Frazier M., and Jawerth B. Decompositions of Besov spaces. Indiana Math. Jour. 34 (1985) 777-799
99. Frazier M., and Jawerth B. The -transform and applications to distribution spaces. Function spaces and Applications (Lund, 1986). Lecture Notes in math (1988), Springer, Berlin, New York 1302
100. Frazier M., and Jawerth B. The -transform and applications to distribution spaces. Function spaces and Applications (Lund, 1986). Lecture Notes in math (1988), Springer, Berlin, New York 233-246
101. Frazier M., and Jawerth B. A discrete transform and decompositions of distribution spaces. Jour. Functional Anal. 93 (1990) 34-170
102. p. Bull. AMS 13 (1985) 1-21
103. Gabardo, J. P. and Han, D. Subspace Weyl-Heisenberg frames, preprint.
104. Gabardo, J. P. and Han, D., private communication.
105. Gabor D. Theory of communications. Jour. Inst. Elec. Eng. (London) 93 (1946) 429-457
106. Gohberg I., and Goldberg S. Basic Operator Theory (1981), Birkhauser, Boston
107. Goupillaud P., Grossmann A., and Morlet J. Cycle-octave and related transforms in seismic signal analysis. Geoexploraation 23 (1985/85) 85-102
108. Govil, N. K. and Zalik, R. A. Perturbation of the Haar wavelet. Proc. Amer. Math. Soc., to appear.
109. n using the Gabor frames, Jour. Four. Anal. and Appl., to appear.
110. Gröchenig K. Describing functions: Atomic decompositions versus frame. Monatshefte fur Math. 112 (1991) 1-41
111. to appear. Gröchenig K. Foundations of time-frequency analysis (2000), Birkhauser
112. Grossmann A. Wavelet transforms and edge detection. In: Albeverio S., et al. (Ed). Stochastic Processing in Physics and Engineering (1988), D. Reidel, Dordrecht, the Netherlands 149-157
113. Grossmann A., and Morlet J. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15 (1984) 723-736
114. Grossmann A., Morlet J., and Paul T. Transforms associated to square integrable group representations, I. General results. Jour. Math. Physics 26 (1985) 2473-2479
115. Halmos P.R. A Hilbert Space Problem Book (1967), Van Nostrand, Princeton
116. Han, D. and Larson, D. Frames, Bases and Group Representations, to appear, Memoris AMS.
117. Heil C. Generalized harmonic analysis in higher dimensions: Weyl-Heisenberg frames and the Zak transform. Ph.D. Thesis (1990), University of Maryland, College Park, MD
118. Heil C., Ramanathan J., and Topiwala P. Linear independence of time-frequency shifts. Proc. AMS 124 9 (1996) 2787-2795
119. Heil C., and Walnut D. Continuous and discrete wavelet transforms. SIAM Review 31 4 (1989) 628-666
120. Hilding S.H. Note on completeness theorems of Paley-Wiener type. Annals of Math. 49 (1948) 953-955
121. Holub J.R. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. Proceedings AMS 122 (1994) 779-785 3
122. Janssen A.J.E.M. Bergmann transform, Zak transform, and coherent states. J. Math. Phys. 23 (1982) 720-731
123. Janssen A.J.E.M. The Zak transform: A signal transform for sampled time-continuous signals. Phillips J. Res. 43 (1988) 720-731
124. Janssen A.J.E.M. Signal analytic proofs of two basic results on lattice expansions. Appl. Comp. Har. Anal. 1 4 (1994) 350-354
125. Janssen A.J.E.M. Duality and biorthogonality for Weyl-Heisenberg frames. Jour. Four Anal. and Appls. 3 No. 5 (1995) 583-596
126. Janssen A.J.E.M. The duality condition for Weyl-Heisenberg frames. In: Feichtinger H.G., and Strohmer T. (Eds). Gabor Analysis and Algorithms: Theory and Applications (1998), Birkhauser, Boston 33-84
127. Kalton N.J., Peck N.T., and Roberts J.W. An F-space sampler. London Mathematical Society Lecture Notes 89 (1984), Cambridge University Press
128. Kato T. Perturbation Theory for Linear Operators (1976), Springer, New York
129. Katznelson Y. An Introduction to Harmonic Analysis (1968), John Wiley, New York
130. Kim, H. O. and Lim, J. K. Frame multiresolution analysis, preprint.
131. Kim, H. O. and Lim, J. K. New characterizations of Riesz bases, preprint.
132. Kohn W. Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115 (1959) 809-821
133. *-modules (A toolkit for operator algebrasts) (1995), Cambridge University Press, Cambridge
134. Li S. On general frame decompositions. Numer. Funct. Anal. and Optimiz 16 (1995) 1181-1191
135. Lindenstrauss J., and Tzafriri L. Classical Banach spaces 1: Sequence spaces. Ergebnisse der Mathematik und ihrere Grenzgebiete (1977), Springer-Verlag 92
136. Linnel, P. Von Neumann algebras and linear independence of translates, Proc. AMS, to appear.
137. Littlewood J.E. Some Problems in Real and Complex Analysis. Health Mathematical Monographs, Lexington, Massachusetts (1968)
138. Low F. Complete sets of wave packets. A passion for Physics-Essays in honor of Geoffrey Chew (1985), World Scientific, Singapore 17-22
139. Lyubarski Y.I. Frames in the Bargmann space of entire functions. Adv. Soviet Math. 429 (1992) 107-113
140. Lyubarski Y.L., and Seip K. Sampling and interpolation sequences for multiband-limited functions and exponential bases on disconnected sets. Jour. Fourier Anal. and Appls. 3 5 (1997) 597-615
141. Lyubarski Y.L., and Seip K. Convergence and summability of Gabor expansion at the Nyquist density. Jour. Fourier Anal. Appls 5 2, 3 (1999) 127-157
142. 2(ℝ). Transaction AMS 315(1) 69-87
143. Mallat S. A Wavelet Tour of Signal Processing (1998), Academic Press, New York
144. Meyer Y. Principle d'incetitude, bases hilbertiennes et algébres d'operateurs. Séminaire Bourbaki 662 (1985/86)
145. Meyer Y. Ondelettes et functions splines. Séminaire EDP. Ecole Polytechnique, Paris (1986)
146. Meyer Y. Wavelets with compact support. Zygmund Lectures (1987), University of Chicago
147. Neumann C. Untersuchungen uber das logarithmische und Newtonsche potential (1877), Teubner, Leipzig
148. Odlyzko A., and Poonen B. Zeroes of polynomials with 0, 1 coefficients. Ens. Math. 39 (1993) 317-348
149. Olson T.E., and Zalik R.A. Nonexistence of a Riesz basis of translates. Approx. In: Anastassiou G.A. (Ed). Theory, proceedings of the Sixth Southeastern Approximation Theorists Annual Conference (1992), Marcel Dekker, New York
150. Paley R.E.A.C., and Wiener N. Fourier transforms in the complex domain. AMS Colloquim Publications 19 (1934)
151. Rado T., and Reichelderfer P. Continuous Transformation in Analysis (1955), Springer-Verlag, Berlin, New York
152. *-algebras. Amer. Math. Soc., Providence, R.I. (1998)
153. Ramanthan J., and Steger T. Incompleteness of sparse coherent states. Appl. comp. Harn. Anal. 2 (1995) 148-153
154. preprint. Rehr J.J., Sullivan D.J., Wilkins J.W., and Wilson K.G. Phase space Wannier functions in electronic structure calculations (1987), Cornell University
155. Rieffel M.A. Neumann algebras associated with pairs of lattices in Lie groups. Math. Anal. 257 (1981) 403-418
156. *-algebras. Advances in Math. 13 (1974) 176-257
157. Rieffel M.A. Projection modules over higher-dimensional noncommutative tori. Canad J. Math. 40 No. 2 (1988) 257-338
158. d). Duke Math. J. 89 (1997) 237-282
159. d). Canadian J. of Math. 47 5 (1995) 1051-1094
160. Schempp W. Radar ambituity functions, the Heisenberg group, and holomorphic theta series. Proc. AMS 92 (1984) 103-110
161. Scherzer, O. and Strohmer, T. A multi-level algorithm for the solution of moment problems, preprint.
162. 2-π, π). Jour. of Functional Analysis 130 1 (1995) 131-160
163. Seip K., and Wallsten R. Density theorems for sampling and interpolation in the Bergmann-Fock space II. J. Reine Angewandtz Multematik 429 (1992) 107-113
164. Shannon C.E. Communications in the presence of noise. Proc. IRE 37 (1949) 10-21
165. Strohmer T. Numberical algorithms for discrete Gabor expansions. In: Feichtinger H.G., and Strohmer T. (Eds). Gabor analysis theory and applications (1997), Birkhäuser
166. Strohmer, T. Rates of convergence for the approximation of dual shift-invariant systems, preprint.
167. Vershynin, R. Frames and contact points of convex bodies, preprint.
168. Walnut D. Weyl-Heisenberg wavelet expansions: Existence and stability in weighted spaces. Ph.D. thesis (1989), University of Maryland, College Part, MD
169. Wiener N. The Fourier Integral and Certain of Its Applications (1933), MIT Press, Cambridge
170. Wexler J., and Raz S. Discrete Gabor expansions. Signal Processing 21 (1990) 207-220
171. Whittaker J.M. Interpolation Function Theory, in Cambridge Tracts in Mathematics and Mathematical Physics (1935), Cambridge University Press, Cambridge
172. preprint. Wilson K.G. Generalized Wannier functions (1987), Cornell University
173. The Wutam Consortium, Basic properties of wavelets. Jour. of Fourier Anal. and Appls. to appear
174. Young R. An Introduction to Nonharmonic Fourier Series (1980), Academic Press, New York
175. Zak J. Finite translations in solid state. Physics Phys. Rev. Lett. 19 (1967) 1385-1397
176. Zwaan M. Approximation of the solution to the moment problem in a Hilbert space. Number. Funct. Anal. Optim. 11 (1990) 601-608
177. Zygmund A. Trigonometric Series Vol. I (1959), Cambridge University Press, Cambridge