Normalized tight vs. general frames in sampling problems. (English) Zbl 1370.42025

Košir, Tomaž; Omladič, Matjaž

Adv. Oper. Theory 2, No. 2, 114-125 (2017).

Summary: We present a new approach to sampling theory using the operator theory framework. We use a replacement operator and replace general frames of the sampling and reconstruction subspaces by normalized tight frames. The replacement can be done in a numerically stable and efficient way. The approach enables us to unify the standard consistent reconstruction results with the results for quasiconsistent reconstruction. Our approach naturally generalizes to consistent and quasiconsistent reconstructions from several samples. Not only we can handle sampling problems in a more efficient way, we also answer questions that seem to be open so far.

MSC:

42C15	General harmonic expansions, frames
94A20	Sampling theory in information and communication theory
15A30	Algebraic systems of matrices
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords:

sampling theory; consistent and quasiconsistent reconstructions; frames and normalized tight frames; replacement operator; several samples

Software:

mftoolbox

References:

[1]	J. G. Aiken, J. A. Erdos, and J. A. Goldstein, On L”owdin orthogonalization, Internat. J. Quantum Chem. 18 (1980), 1101–1108.
[2]	S. T. Ali, J. P. Antoine, and J. P. Gazeau, Continuous frames in Hilbert space, Ann. Physics 222 (1993), no. 1, 1–37. · Zbl 0782.47019
[3]	J. Antezana and G. Corach, Sampling Theory, oblique projections and a question by Smale and Zhou, Appl. Comput. Harmon. Anal. 21 (2006), 245–253.
[4]	M. L. Arias and C. Conde, Generalized inverses and sampling problems, J. Math. Anal. Appl. 398 (2013), 744–751. · Zbl 1254.42036
[5]	R. Balan, Equivalence relations and distances between Hilbert frames, Proc. Amer. Math. Soc. 127 (8) (1999), 2353–2366. · Zbl 0922.42025
[6]	A. B”ottcher and I. M. Spitkovsky, A gentle guide to the basics of two projections theory, Lin. Algebra Appl. 432 (2010), 1412–1459. · Zbl 1189.47073
[7]	A. L. Cauchy, Memoire sur diverses formules d’analyser, Comptes Rendues 12 (1841), 283–298.
[8]	G. Corach and J. I. Giribet, Oblique projections and sampling problems, Integral Equations Operator Theory 70 (2011), 307–322. · Zbl 1248.94041
[9]	T. G. Dvorkind and Y. C. Eldar, Robust and consistent sampling, IEEE Signal Processing letters 16 (2009), no. 9, 739–742.
[10]	Y. C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier Anal. Appl. 9 (2003), 77–99. · Zbl 1028.94020
[11]	Y. C. Eldar and T. G. Dvorkind, A minimum squared-error framework for generalized sampling, IEEE Trans. Signal Process. 54 (2006), no. 6, 2155–2167. · Zbl 1374.94720
[12]	Y. C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Wavelets Multiresolut. Inf. Process. 3 (2005), no. 4, 347–359. · Zbl 1075.42010
[13]	P. A. Fillmore and J. P. Williams, On operator ranges, Adv. Math. 7 (1971), 254–281. · Zbl 0224.47009
[14]	M. Frank, V. I. Paulsen, and T. R. Tiballi, Symmetric approximation of frames and bases in Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002), no. 2, 777–793. · Zbl 0984.42021
[15]	P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. · Zbl 0187.05503
[16]	D. Han and D. R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147, no. 697, 2000. · Zbl 0971.42023
[17]	N. J. Higham, Functions of matrices. Theory and computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. · Zbl 1167.15001
[18]	N. J. Higham and R. S. Schreiber, Fast polar decomposition of an arbitrary matrix, SIAM J. Sci. Statist. Comput. 11 (1990), 648–655. FRAMES IN SAMPLING PROBLEMS125 · Zbl 0711.65024
[19]	A. Hirabayashi and M. Unser, Consistent Sampling and Signal Recovery, IEEE Trans. Signal Process. 55 (2007), no. 8, 4104–4115. · Zbl 1390.94652
[20]	M. Hladnik and M. Omladic, Spectrum of the product of operators, Proc. Amer. Math. Soc. 102 (1988), 300–302. · Zbl 0661.47005
[21]	V. A. Kotel’nikov,On the transmission capacity of ether and wire in electrocommunications, in: First All-Union Conf. Questions of Commun., 1933. (English Translation in Appl. Numer. Harmon. Anal., Modern sampling theory, 27-45, Birkh”auser, Boston,
[22]	)
[23]	M. G. Krein, M. A. Krasnoselski, and D. P. Milman, On the defect numbers of operators in Banach spaces and on some geometric questions, Trudy Inst. Mat. Akad. Nauk Ukrain. SSR, 11 (1948), 97–112.
[24]	D. R. Larson, Frames and wavelets from an operator-theoretic point of view, In Operator algebras and operator theory (Shanghai, 1997), 201–218, Contemp. Math. 228, Amer. Math. Soc., Providence, RI, 1998. · Zbl 0939.42024
[25]	M. Omladic, Spectra of the difference and product of projections, Proc. Amer. Math. Soc. 99 (1987), 317–318.
[26]	C. E. Shannon, Communication in the presence of noise, Proc. Inst. Radio Eng. 37 (1949), no. 1, 10–21. (Reprint as classic paper in: Proc. IEEE 86 (2) (1998).)
[27]	T. R. Tiballi, Symmetric orthogonalization of vectors in Hilbert spaces. Ph.D. Thesis, University of Houston, Houston, Texas, USA, 1991.
[28]	M. Unser, Sampling 50 Years After Shannon, IEEE Proc. 88 (2000), 569–587.
[29]	M. Unser and A. Aldroubi, A general sampling theory for nonideal acquisition devices, IEEE Trans. Signal Process. 42 (11) (1994), 2915–2925.
[30]	E. T. Whittaker, On the functions which are represented by the expansions of the interpolation-theory, Proc. Roy. Soc. Edinburgh 35 (1915), 181–194. 1 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia; Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia; and Institute Jozef Stefan, Jamova 39, 1000 Ljubljana, Slovenia. E-mail address:[e-mail] E-mail address:[e-mail] 1. Introduction2. Preliminaries 2.1. Frames2.2. Projections3. Normalized tight frames 4. Reconstructions for two or more samples for normalized tight framesReferences