Abstract
In the study of vector spaces, one of the most important concepts is that of a basis. A basis provides us with an expansion of all vectors in terms of “elementary building blocks” and hereby helps us by reducing many questions concerning general vectors to similar questions concerning only the basis elements. However, the conditions to a basis are very restrictive – no linear dependence between the elements is possible, and sometimes we even want the elements to be orthogonal with respect to an inner product. This makes it hard or even impossible to find bases satisfying extra conditions, and this is the reason that one might look for a more flexible tool.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alexeev, B., Cahill, J., Mixon, D.G.: Full spark frames. J. Fourier Anal. Appl. 18(6), 1167–1194 (2012)
Balan, R., Bodmann, B., Casazza, P., Eddin, D.: Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15(4), 488–501 (2009)
Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20, 345–356 (2006)
Balan, R., Wang, Y.: Invertibility and robustness of phaseless reconstruction. Appl. Comput. Harmon. Anal. 38, 469–488 (2015)
Benedetto, J., and Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18(2–4), 357–385 (2003)
Benedetto, J., Powell A., Yilmaz, Ö.: Second order sigma-delta quantization of finite frame expansions. Appl. Comput. Harmon. Anal. 20, 126–148 (2006)
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)
Blum, J., Lammers, M., Powell, A.M., Yilmaz, Ö.: Errata to: Sobolev duals in frame theory and sigma-delta quantization. J. Fourier Anal. Appl. 16, 382–382 (2010)
Bodmann, B.G.: Frames as codes. In: Kutyniok, G., Casazza, P.G. (eds.) Finite Frames. Applied and Numerical Harmonic Analysis, pp. 241–266. Birkhäuser, New York (2013)
Bodmann, B.G., Casazza, P.G.; Kutyniok, G.: A quantitative notion of redundancy for finite frames. Appl. Comput. Harmon. Anal. 30(3), 348–362 (2011)
Bownik, M., Speegle, D.: Linear independence of Parseval wavelets. Ill. J. Math. 54, 771–785 (2010)
Casazza, P. G., Fickus, M.,, Heinecke, A., Wang, Y., Zhou, Z.: Spectral tetris fusion frame constructions. J. Fourier Anal. Appl. 18(4), 828–851 (2012)
Casazza, P.G., Fickus, M., Mixon, D., Wang, Y., Zhou, Z.: Constructing tight fusion frames. Appl. Comput. Harmon. Anal. 30, 175–187 (2011)
Casazza, P.G., Kutyniok, G. (eds): Finite Frames: Theory and Applications. Birkhäuser, Boston (2012)
Casazza, P.G., Kutyniok, G.: Frames and subspaces. In: Wavelets, Frames, and Operator Theory. Contemporary Mathematics, vol. 345, pp. 87–113. American Mathematical Society, Providence (2004)
Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25, 114–132 (2008)
Casazza, P.G., Leonhard, N.: Classes of finite equal norm Parseval frames. Contemp. Math. 451, 11–31 (2008)
Chen, S., Donoho, D., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–157 (2001)
Christensen, O., Lindner, A.: Decompositions of wavelets and Riesz frames into a finite number of linearly independent sets. Linear Algebra Appl. 355, 147–159 (2002)
Goyal, V.K., Kovačević, J., Kelner, A.J.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10(3), 203–233 (2000)
Gröchenig, K.: Acceleration of the frame algorithm. IEEE Trans. Signal Process. 41(12), 3331–3340 (1993)
Han, D., Kornelson, K., Larson, D., Weber, E.: Frames for undergraduates. Student Mathematical Library, vol. 40. American Mathematical Society, Providence (2007)
Heil, C., Ramanathan, J., Topiwala, P.: Linear independence of time-frequency translates. Proc. Am. Math. Soc. 124, 2787–2795 (1996)
Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004)
Jasper, J., Mixon, D.G., Fickus, M.: Kirkman equiangular tight frames and codes. IEEE Trans. Inf. Theory 60(1), 170–181 (2014)
Lammers, M., Powell, A., Yilmaz, “O.: Alternative dual frames for digital-to-analog conversion in sigma-delta quantization. Adv. Comput. Math. 32, 73–102 (2010)
Linnell, P.: Von Neumann algebras and linear independence of translates. Proc. Am. Math. Soc. 127(11), 3269–3277 (1999)
Mixon, D.G., Quinn, C.J., Kiyavash, N., Fickus, M.: Fingerprinting with equiangular tight frames. IEEE Trans. Inf. Theory 59(3), 1855–1865 (2013)
Powell, A.M., Saab, R., Yilmaz, Ö: Quantization and finite frames. In: Casazza, P., Kutyniok, G. (eds.) Finite Frames, chap. 8 Birkhäuser, Boston (2012)
Strohmer, T.: A note on equiangular tight frames. Linear Algebra Appl. 429(1), 326–330 (2008)
Strohmer, T., Heath, R.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)
Sustik, M.A., Tropp, J.A., Dhillon, I.S., Heath, R.W., Jr.: On the existence of equiangular tight frames. Linear Algebra Appl. 426(2–3), 619–635 (2007)
Tropp, J.A., Dhillon, I.S.; Heath, R.W., Jr., Strohmer, T.: Designing structured tight frames via an alternating projection method. IEEE Trans. Inf. Theory 51(1), 188–209 (2005)
Xia, P., Zhou, S., Giannakis, S.B.: Achieving the Welch Bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005)
Zimmermann, G.: Normalized tight frames in finite dimensions. In: Jetter, K., Haußmann, W., Reimer, M. (eds.) Recent Progress in Multivariate Approximation, pp. 249–252. Birkhäuser, Boston (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Christensen, O. (2016). Frames in Finite-Dimensional Inner Product Spaces. In: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-25613-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-25613-9_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-25611-5
Online ISBN: 978-3-319-25613-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)