Abstract
Inspired by an elegant theorem of Boas and Pollard (and related results by Kazarian, Price, Talalyan, Zink, and others), we discuss multiplicative completion of redundant systems in Hilbert and Banach function spaces.
This work was partially supported by a grant from the Simons Foundation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Aldroubi, K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001)
R. Balan, P.G. Casazza, C. Heil, Z. Landau, Density, overcompleteness, and localization of frames, I. Theory. J. Fourier Anal. Appl. 12, 105–143 (2006)
R. Balan, P.G. Casazza, C. Heil, Z. Landau, Density, overcompleteness, and localization of frames, II. Gabor frames. J. Fourier Anal. Appl. 12, 307–344 (2006)
J.J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389–427 (1998)
S. Bishop, Comparison theorems for separable wavelet frames. J. Approx. Theory 161, 432–447 (2009)
R.P. Boas, H. Pollard, The multiplicative completion of sets of functions. Bull. Am. Math. Soc. 54, 518–522 (1948)
J.S. Byrnes, Functions which multiply bases. Bull. Lond. Math. Soc. 4, 330–332 (1972)
J.S. Byrnes, Complete multipliers. Trans. Am. Math. Soc. 172, 399–403 (1972)
J.S. Byrnes, D.J. Newman, Completeness preserving multipliers. Proc. Am. Math. Soc. 21, 445–450 (1969)
O. Christensen, An Introduction to Frames and Riesz Bases (Birkhäuser, Boston, 2003)
I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)
K. Gröchenig, H. Razafinjatovo, On Landau’s necessary density conditions for sampling and interpolation of band-limited functions. J. Lond. Math. Soc. 54(2), 557–565 (1996)
C. Heil, Wiener amalgam spaces in generalized harmonic analysis and wavelet theory, Ph.D. thesis, University of Maryland, College Park, 1990
C. Heil, A Basis Theory Primer, Expanded Edition (Birkhäuser, Boston, 2011)
C. Heil, G. Kutyniok, The homogeneous approximation property for wavelet frames. J. Approx. Theory 147, 28–46 (2007)
C. Heil, G. Kutyniok, Density of frames and Schauder bases of windowed exponentials. Houston J. Math. 34, 565–600 (2008)
R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973)
K.S. Kazarian, The multiplicative completion of basic sequences in L p,  1 ≤ p < ∞, to bases in L p (Russian). Akad. Nauk Armjan. SSR Dokl. 62, 203–209 (1976)
K.S. Kazarian, The multiplicative complementation of some incomplete orthonormal systems to bases in L p, 1 ≤ p < ∞ (Russian). Anal. Math. 4, 37–52 (1978)
K.S. Kazarjan, Summability of generalized Fourier series and Dirichlet’s problem in L p(dμ) and weighted H p-spaces (p > 1). Anal. Math. 13, 173–197 (1987)
K.S. Kazarian, R.E. Zink, Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in L 2. Trans. Am. Math. Soc. 349, 4367–4383 (1997)
J.J. Price, Topics in orthogonal functions. Am. Math. Monthly 82, 594–609 (1975)
J.J. Price, R.E. Zink, On sets of functions that can be multiplicatively completed. Ann. Math. 82, 139–145 (1965)
J. Ramanathan, T. Steger, Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2, 148–153 (1995)
H. Reiter, Classical Harmonic Analysis and Locally Compact Groups (Oxford University Press, Oxford, 1968)
W. Sun, X. Zhou, Density of irregular wavelet frames. Proc. Am. Math. Soc. 132, 2377–2387 (2004)
A.A. Talalyan, On the convergence almost everywhere of subsequences of partial sums of general orthogonal series. Izv. Akad. Nauk Armyan SSR Ser. Fiz.-Mat. 10, 17–34 (1957)
G.J. Yoon, C. Heil, Duals of weighted exponentials. Acta Appl. Math. 119, 97–112 (2012)
R. Young, An Introduction to Nonharmonic Fourier Series, Revised First Edition (Academic, San Diego, 2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Heil, C. (2021). Reflections on a Theorem of Boas and Pollard. In: Rassias, M.T. (eds) Harmonic Analysis and Applications. Springer Optimization and Its Applications, vol 168. Springer, Cham. https://doi.org/10.1007/978-3-030-61887-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-61887-2_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-61886-5
Online ISBN: 978-3-030-61887-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)