SpringerLink
Log in

Duality for Frames

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Often in the literature the squares of those numbers are called lower and upper frame bounds.

  2. Consider, e.g., \(\ell _2(\mathbb {N})\) with the standard unit vector basis \(\mathcal {O}=\{e_n\}_{n\in \mathbb {N}}\). Then \(X=\{ne_n\}_{n\in \mathbb {N}}\) satisfies (3.5) but is not a Bessel sequence.

  3. Note that one might also consider complex conjugations of the entries of the pre-Gramian in (3.10) without introducing essential changes to the discussion that follows.

  4. The factor \(2^d\) is an artifact of the dyadic dilations we use. The construction in [73] works for general dilation matrices and \(2^d\) is being replaced by the determinant of the dilation matrix.

References

  1. Bao, C., Ji, H., Shen, Z.: Convergence analysis for iterative data-driven tight frame construction scheme. Appl. Comput. Harm. Anal. 38(3), 510–523 (2015)

    Article  MathSciNet  Google Scholar 

  2. Blum, J., Lammers, M., Powell, A.M., Yilmaz, Ö.: Sobolev duals in frame theory and sigma-delta quantization. J. Fourier Anal. Appl. 16, 365–381 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Equivalence of DFT filter banks and Gabor expansions. In: SPIE 95, Wavelet Applications in Signal and Image Processing III, vol. 2569, pp. 128–139, San Diego (1995)

  4. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Oversampled FIR and IIR DFT filter banks and Weyl–Heisenberg frames. In: Proceedings of the IEEE ICASSP-96, vol. 3, pp. 1391–1394, Atlanta (GA) (1996)

  5. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Proces. 46(12), 3256–3268 (1998)

    Article  Google Scholar 

  6. Bownik, M., Lemvig, J.: The canonical and alternate duals of a wavelet frame. Appl. Comput. Harm. Anal. 23(2), 263–272 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cahill, J., Fickus, M., Mixon, D.G., Poteet, M.J., Strawn, N.: Constructing finite frames of a given spectrum and set of lengths. Appl. Comp. Harm. Anal. 35, 52–73 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cai, J.F., Chan, R.H., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comp. Harm. Anal. 24(2), 131–149 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cai, J.F., Dong, B., Osher, S., Shen, Z.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cai, J.F., Dong, B., Shen, Z.: Image restoration: a wavelet frame based model for piecewise smooth functions and beyond. Preprint

  11. Cai, J.F., Ji, H., Liu, C., Shen, Z.: Blind motion deblurring from a single image using sparse approximation. In: IEEE CVPR 2009, pp. 104–111. IEEE (2009)

  12. Cai, J.F., Ji, H., Liu, C.J., Shen, Z.: Blind motion deblurring using multiple images. J. Comput. Phys. 228(14), 5057–5071 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cai, J.F., Ji, H., Shen, Z., Ye, G.B.: Data-driven tight frame construction and image denoising. Appl. Comput. Harm. Anal. 37(1), 89–105 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cai, J.F., Osher, S., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imag. Sci. 2(1), 226–252 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cai, J.F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)

    Article  MathSciNet  Google Scholar 

  16. Casazza, P.G., Christensen, O.: Frames containing a Riesz basis and preservation of this property under perturbations. SIAM J. Math. Anal. 29(1), 266–278 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Casazza, P.G., Fickus, M., Mixon, D.G., Wang, Y., Zhou, Z.: Constructing tight fusion frames. Appl. Comput. Harm. Anal. 30, 175–187 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Casazza, P.G., Kutyniok, G.: Finite Frames: Theory and Applications. Birkhäuser, Boston (2012)

    Google Scholar 

  19. Casazza, P.G., Kutyniok, G., Lammers, M.C.: Duality principles in frame theory. J. Fourier Anal. Appl. 10, 383–408 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Casazza, P.G., Kutyniok, G., Lammers, M.C.: Duality principles, localization of frames, and Gabor theory. Proc. SPIE Wavelets XI 5914, 389–397 (2005)

    Google Scholar 

  21. Chai, A., Shen, Z.: Deconvolution: a wavelet frame approach. Numer. Math. 106, 529–587 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Chen, D.R., Han, B., Riemenschneider, D.: Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments. Adv. Comput. Math. 13, 131–165 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Christensen, L.H., Christensen, O.: Construction of smooth compactly supported windows generating dual pairs of Gabor frames. Asian-Eur. J. Math. 6 (2013)

  24. Christensen, O.: Pairs of dual Gabor frame generators with compact support and desired frequency localization. Appl. Comput. Harm. Anal. 20, 403–410 (2006)

    Article  MATH  Google Scholar 

  25. Christensen, O.: Frames and Bases: An Introductory Course. Birkhäuser, Boston (2008)

    Book  Google Scholar 

  26. Christensen, O., Favierand, S., Zó, F.: Irregular wavelet frames and Gabor frames. Approx. Theory Appl. 17(3), 90–101 (2001)

    MATH  MathSciNet  Google Scholar 

  27. Christensen, O., Kim, H.O., Kim, R.Y.: On entire functions restricted to intervals, partition of unities, and dual Gabor frames. Appl. Comput. Harm. Anal. 38(1), 72–86 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  28. Christensen, O., Kim, H.O., Kim, R.Y.: On the duality principle by Casazza, Kutyniok, and Lammers. J. Fourier Anal. Appl. 17, 640–655 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Christensen, O., Kim, R.Y.: Pairs of explicitly given dual Gabor frames in \(L_2({\mathbb{R}}^d)\). J. Fourier Anal. Appl. 12, 243–255 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Christensen, O., **ao, X.Z., Zhu, Y.C.: Characterizing R-duality in Banach spaces. Acta Math. Sinica (Engl. Ser.) 29(1), 75–84 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  32. Coifman, R.R., Donoho, D.L.: Translation-Invariant De-Noising. Springer, Berlin (1995)

    Book  Google Scholar 

  33. Cvetkovic̀, Z., Vetterli, M.: Oversampled filter banks. IEEE Trans. Signal Process 46(5), 1245–1255 (1998)

    Article  MathSciNet  Google Scholar 

  34. Cvetkovic̀, Z., Vetterli, M.: Tight Weyl–Heisenberg frames in \(\ell _2(\mathbb{Z})\). IEEE Trans. Signal Process 46, 1256–1259 (1998)

    Article  MathSciNet  Google Scholar 

  35. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41(7), 909–996 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  36. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 39, 961–1005 (1990)

    Article  MathSciNet  Google Scholar 

  37. Daubechies, I.: Ten lectures on wavelets. In: SIAM, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61 (1992)

  38. Daubechies, I., Grossman, A., Meyer, Y.: Painless non-orthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  39. Daubechies, I., Han, B.: The canonical dual frame of a wavelet frame. Appl. Comput. Harm. Anal. 22, 269–285 (2002)

    Article  MathSciNet  Google Scholar 

  40. Daubechies, I., Han, B.: Pairs of dual wavelet frames from any two refinable functions. Constr. Approx. 20, 325–352 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  41. Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comp. Harm. Anal. 14, 1–46 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  42. Daubechies, I., Jaffard, S., Journé, J.L.: A simple Wilson orthonormal basis with exponential decay. SIAM J. Math. Anal. 22, 554–573 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  43. Daubechies, I., Landau, H.J., Landau, Z.: Gabor time-frequency lattices and the Wexler–Raz identity. J. Fourier Anal. Appl. 1, 437–478 (1994)

    Article  MathSciNet  Google Scholar 

  44. Dong, B., Chien, A., Shen, Z.: Frame based segmentation for medical images. Commun. Math. Sci. 9(2), 551–559 (2010)

    MathSciNet  Google Scholar 

  45. Dong, B., Ji, H., Li, J., Shen, Z., Xu, Y.: Wavelet frame based blind image inpainting. Appl. Comp. Harm. Anal. 32(2), 268–279 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  46. Dong, B., Jiang, Q., Shen, Z.: Image restoration: wavelet frame shrinkage, nonlinear evolution PDEs, and beyond. UCLA CAM Report, vol.13 (2013)

  47. Dong, B., Li, J., Shen, Z.: X-ray CT image reconstruction via wavelet frame based regularization and Radon domain inpainting. J. Sci. Comput. 54(2–3), 333–349 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  48. Dong, B., Shen, Z.: Pseudo-splines, wavelets and framelets. Appl. Comp. Harm. Anal. 11, 78–104 (2007)

    Article  MathSciNet  Google Scholar 

  49. Dong, B., Shen, Z.: MRA-based wavelet frames and applications. IAS Lecture Notes Series, Summer Program on The Mathematics of Image Processing, Park City Mathematics Institute, vol. 19 (2010)

  50. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  51. Dyn, N., Levine, D., Gregory, J.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Gr. 9, 160–169 (1990)

    Article  MATH  Google Scholar 

  52. Fan, Z., Ji, H., Shen, Z.: Dual Gramian analysis: duality principle and unitary extension principle. Math. Comput. (to appear)

  53. Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86(2), 307–340 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  54. Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. Part II. Monatsh. Math. 108(2–3), 129–148 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  55. Feichtinger, H.G., Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146, 464–495 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  56. Feichtinger, H.G., Kozek, W.: Gabor Analysis and Algorithms: Theory and Applications, Chapter Quantization of TF-Lattice Invariant Operators on Elementary LCA Groups, pp. 233–266. Birkhäuser, Boston (1998)

    Book  Google Scholar 

  57. Feichtinger, H.G., Luef, F.: Wiener amalgam spaces for the fundamental identity of Gabor analysis. Collect. Math. 57, 233–253 (2006)

    MathSciNet  Google Scholar 

  58. Feichtinger, H.G., Zimmermann, G.: Gabor Analysis and Algorithms: Theory and Applications, Chapter A Banach Space of Test Functions for Gabor Analysis, pp. 123–170. Birkhäuser, Boston (1998)

    Book  Google Scholar 

  59. Goh, S.S., Jiang, Q.T., **a, T.: Construction of biorthogonal multiwavelets using the lifting scheme. Appl. Comput. Harm. Anal. 9, 336–352 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  60. Gong, Z., Shen, Z., Toh, K.C.: Image restoration with mixed or unknown noises. Multiscale Model. Simul. 12(2), 458–487 (2014)

    Article  Google Scholar 

  61. Gröchenig, K.: Irregular sampling of wavelet and short-time Fourier transforms. Constr. Approx. 9(2–3), 283–297 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  62. Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001)

    Google Scholar 

  63. Gröchenig, K.: The mystery of Gabor frames. J. Fourier Anal. Appl. 20, 865–895 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  64. Han, B.: Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. J. Comput. Appl. Math. 155(1), 43–67 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  65. Han, B.: The projection method for multidimensional framelet and wavelet analysis. Math. Model. Nat. Phenom. 9(5), 83–110 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  66. Han, B.: Algorithm for constructing symmetric dual framelet filter banks. Math. Comput. 84, 767–801 (2015)

    Article  MATH  Google Scholar 

  67. Han, B., Shen, Z.: Dual wavelet frames and Riesz bases in Sobolev spaces. Constr. Approx. 29(3), 369–406 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  68. Herley, C., Vetterli, M.: Wavelets and recursive filter banks. IEEE Trans. Signal Proces. 41(8), 2536–2556 (1993)

    Article  MATH  Google Scholar 

  69. Holschneider, M., Kronland-Martinet, R., Morlet, J., Tchamitchian, P.: A real-time algorithm for signal analysis with the help of the wavelet transform. In: Combes, J.M., Grossmann, A., Tchamitchian, P. (eds.) Wavelets, Time-Frequency Methods and Phase Space, pp. 286–297. Springer, Berlin (1990)

  70. Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104(1), 93–140 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  71. Hou, L., Ji, H., Shen, Z.: Recovering over-/underexposed regions in photographs. SIAM J. Imag. Sci. 6(4), 2213–2235 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  72. Janssen, A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. J. Fourier Anal. Appl. 1, 403–436 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  73. Ji, H., Riemenschneider, S.D., Shen, Z.: Multivariate compactly supported fundamental refinable functions, dual and biorthogonal wavelets. Stud. Appl. Math. 102, 173–204 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  74. Ji, H., Shen, Z.: Compactly supported (bi)orthogonal wavelets by interpolatory refinable functions. Adv. Comput. Math. 11, 81–104 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  75. Ji, H., Shen, Z., Zhao, Y.: Directional frames for image recovery: multi-scale finite discrete Gabor frames. (Preprint)

  76. Jia, R.Q., Shen, Z.: Multiresolution and wavelets. P. Edinb. Math. Soc. 37, 271–300 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  77. Kim, H.O., Kim, R.Y., Lim, J.K., Shen, Z.: A pair of orthogonal frames. J. Approx. Theory 147(2), 196–204 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  78. Kim, I.: Gabor frames with trigonometric spline dual windows. PhD thesis, University of Illinois at Urbana-Champaign (2011)

  79. Kovačević, J., Sweldens, W.: Wavelet families of increasing order in arbitrary dimensions. IEEE Trans. Image Process. 9, 480–496 (2000)

    Article  MATH  Google Scholar 

  80. Landau, H.J.: On the density of phase-space expansions. IEEE Trans. Inf. Theory 39, 1152–1156 (1993)

    Article  MATH  Google Scholar 

  81. Laugensen, R.S.: Gabor dual spline windows. Appl. Comput. Harm. Anal. 27, 180–194 (2009)

    Article  Google Scholar 

  82. Lawton, W., Lee, S.L., Shen, Z.: Convergence of multidimensional cascade algorithm. Numer. Math. 78, 427–438 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  83. Li, M., Fan, Z., Ji, H., Shen, Z.: Wavelet frame based algorithm for 3D reconstruction in electron microscopy. SIAM J. Sci. Comput. 36(1), 45–69 (2014)

    Article  MathSciNet  Google Scholar 

  84. Li, S.: On general frame decompositions. Numer. Funct. Anal. Optim. 16(9–10), 1181–1191 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  85. Liang, J., Li, J., Shen, Z., Zhang, X.: Wavelet frame based color image demosaicing. Inverse Probl. Imag. 7(3), 777–794 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  86. Mallat, S.: A theory of multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. 11, 674–693 (1989)

    Article  MATH  Google Scholar 

  87. Mallat, S.: A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, New York (2008)

    Google Scholar 

  88. Mallat, S., Zhong, S.: Characterization of signals from multiscale edges. IEEE Trans. Pattern Anal. 40(7), 2464–2482 (1992)

    Google Scholar 

  89. Meyer, Y.: Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs. Séminaire Bourbaki 662, 209–223 (1985–1986)

  90. Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge, MA (2012)

    Google Scholar 

  91. Quan, Y., Ji, H., Shen, Z.: Data-driven multi-scale non-local wavelet frame construction and image recovery. J. Sci. Comput. 1–23 (2014)

  92. Riemenschneider, S.D., Shen, Z.: Box splines, cardinal series, and wavelets. In: Chui, C.K. (ed.) Approximation Theory and Functional Analysis, pp. 133–149. Academic Press, New York (1991)

  93. Riemenschneider, S.D., Shen, Z.: Wavelets and pre-wavelets in low dimensions. J. Approx. Theory 71(1), 18–38 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  94. Riemenschneider, S.D., Shen, Z.: Multidimensional interpolatory subdivision schemes. SIAM J. Numer. Anal. 34, 2357–2381 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  95. Riemenschneider, S.D., Shen, Z.: Construction of compactly supported biorthogonal wavelets in \(L_2({\mathbb{R}}^d)\). In: Mastorakis, N.E. (eds.) Physics and Modern Topics in Mechanical and Electrical Engineering, pp. 201–206. World Scientific and Engineering Society Press (1999)

  96. Riemenschneider, S.D., Shen, Z.: Construction of compactly supported biorthogonal wavelets in \(L_2(\mathbb{R}^d)\) II. In: Unser, M.A., Aldroubi, A., Lain, A.F. (eds.) Proceedings of SPIE, vol. 3813, pp. 264–272. Wavelet applications signal and image processing VII (1999)

  97. Ron, A., Shen, Z.: Frames and stable bases for subspaces of \(L{\mathbb{R}}_2(^d)\): the duality principle of Weyl–Heisenberg sets. In: Chu, M., Plemmons, R., Browns, D., Ellison, D. (eds.) Proceedings of the Lanczos Centenary Conference, pp. 422–425. SIAM Pub, Raleigh, NC (1993)

  98. Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces of \(L_2(\mathbb{R}^d)\). Can. J. Math. 47, 1051–1094 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  99. Ron, A., Shen, Z.: Gramian analysis of affine bases and affine frames. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory VIII-Vol2, Wavelets and Multilevel Approximation, pp. 375–382. World Scientific Publishing, New Jersey (1995)

    Google Scholar 

  100. Ron, A., Shen, Z.: Affine systems in \(L_2(\mathbb{R}^d)\) II: dual systems. J. Fourier Anal. Appl. 3, 617–637 (1997)

    Article  MathSciNet  Google Scholar 

  101. Ron, A., Shen, Z.: Affine systems in \(L_2({\mathbb{R}}^d)\): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  102. Ron, A., Shen, Z.: Weyl–Heisenberg frames and Riesz bases in \(L_2({\mathbb{R}}^d)\). Duke Math. J. 89, 237–282 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  103. Ron, A., Shen, Z.: Generalized shift invariant systems. Constr. Approx. 22, 1–45 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  104. Schroeck, F.E.: Quantum Mechanics on Phase Space. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  105. Seip, K.: Density theorems for sampling and interpolation in the Bargmann–Fock space. Bull. Am. Math. Soc. 26(2), 322–328 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  106. Seip, K.: On the connection between exponential bases and certain related sequences in \(L^2(-\pi,\pi )\). J. Funct. Anal. 130, 131–160 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  107. Shen, Z., Toh, K.C., Yun, S.: An accelerated proximal gradient algorithm for frame-based image restoration via the balanced approach. SIAM J. Imag. Sci. 4(2), 573–596 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  108. Shensa, M.J.: The discrete wavelet transform: Wedding the à trous and Mallat algorithms. IEEE Trans. Signal Process. 40(10), 2464–2482 (1992)

    Article  MATH  Google Scholar 

  109. Soman, A.K., Vaidyanathan, P.P.: On orthonormal wavelets and paraunitary filter banks. IEEE Trans. Signal Process. 41(3), 1170–1183 (1993)

    Article  MATH  Google Scholar 

  110. Stahl, D.: Multivariate polynomial interpolation and the lifting scheme with an application to scattered data approximation. PhD thesis, Technische Universität Kaiserslautern (2013)

  111. Sun, W., Zhou, X.: Irregular wavelet/Gabor frames. Appl. Comput. Harm. Anal. 13(1), 63–76 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  112. Sweldens, W.: The lifting scheme: a new philosophy in biorthogonal wavelet construction. In: SPIE’s 1995 International Symposium on Optical Science, Engineering, and Instrumentation, vol. 2569, pp. 68–79 (1995)

  113. Tai, C., Zhang, X., Shen, Z.: Wavelet frame based multiphase image segmentation. SIAM J. Imag. Sci. 6(4), 2521–2546 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  114. Vetterli, M., Herley, C.: Wavelets and filter banks: theory and design. IEEE Trans. Signal Process. 40(9), 2207–2232 (1992)

    Article  MATH  Google Scholar 

  115. Walnut, D.J.: Continuity properties of the Gabor frame operator. J. Math. Anal. Appl. 165(2), 479–504 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  116. Wexler, J., Raz, S.: Discrete Gabor expansions. Signal Process. 21, 207–220 (1990)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by Singapore MOE Research Grants R-146-000-165-112 and R-146-000-178-112. The last author was also supported by the Tan Chin Tuan Centennial Professorship.

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore, 119076, Singapore

    Zhitao Fan, Andreas Heinecke & Zuowei Shen

Authors
  1. Zhitao Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Heinecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zuowei Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Heinecke.

Additional information

Communicated by Peter G. Casazza.

Appendices

Appendix 1: Primary Wavelet Masks of Example 5.9

$$\begin{aligned}&\frac{1}{2} \begin{pmatrix} 0 &{}\quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1&{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 2 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}. \end{aligned}$$

Appendix 2: Primary Wavelet Masks of Example 5.10

$$\begin{aligned}&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{2}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{2} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned}&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4}\begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -1&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix},\\&\frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -1 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \frac{1}{4} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 2 &{} \quad -1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, Z., Heinecke, A. & Shen, Z. Duality for Frames. J Fourier Anal Appl 22, 71–136 (2016). https://doi.org/10.1007/s00041-015-9415-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-015-9415-0

Keywords

Mathematics Subject Classification

Search

Navigation