Abstract
Given a frame F = {f j } for a separable Hilbert space H, we introduce the linear subspace \(H^{p}_{F}\) of H consisting of elements whose frame coefficient sequences belong to the ℓp-space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as \(H^{p}_{F}\)-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in \(H_{F}^{p}\) converges in both the Hilbert space norm and the ||·||F, p-norm which is induced by the ℓp-norm.
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Balan, R., Casazza, P., Heil, C., Landau, Z.: Density, overcompleteness, and localization of frames. I. Theory. J. Fourier Anal. Appl. 12, 105–143 (2006)
Balan, R., Casazza, P., Heil, C., Landau, Z.: Density, overcompleteness, and localization of frames. II. Gabor systems. J. Fourier Anal. Appl. 12, 309–344 (2006)
Benedetto, J.J.: Harmonic Analysis and Applications. CRC Press, Boca Raton, New York, London, Tokyo (1996)
Carothers, N.L.: A Short Course on Banach Space Theory. Cambridge University Press, Cambridge (2005)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)
Cordero, E., Gröchenig, K.: Localization of frames II. Appl. Comput. Harmon. Anal. 17, 29–47 (2004)
Dai, X., Larson, D.R.: Wandering vectors for unitary systems and orthogonal wavelets. Mem. Am. Math. Soc. 134(640), 1–68 (1998)
Daubechies, I.: Ten Lectures on Wavelets. CBS-NSF Regional Conferences in Applied Mathematics No. 61, SIAM (1992)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Fornasier, M., Gröchenig, K.: Intrinsic localization of frames. Constr. Approx. 22(3), 395–415 (2005)
Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10, 105–132 (2004)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Borkhäuser, Boston, Basel, Berlin (2000)
Han, D., Larson, D.R.: Frames, bases and group representations. Mem. Am. Math. Soc. 147(697), 1–94 (2000)
Heil, C., Larson, D.: Operator theory and modulation spaces. Contemp. Math. 451, 137–150 (2008)
Hernández, E., Weiss, G.: A First Course on Wavelets. With a foreword by Yves Meyer. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, (1996)
Jia, R.Q.: Shift-invariant spaces and linear operator equations. Isr. J. Math. 103, 259–288 (1998)
Jia, R.Q., Micchelli, C.A.: Using the refinement equations for the construction of pre-wavelets II: Powers of two. In: Laurent, P.J., Le Mehaute, A., Schumaker, L.L., (eds.) Curves and Surfaces, pp. 209–246. Academic Press, New York (1991)
Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1998)
Sun, Q.: Wiener’s lemma for infinite matrices. Trans. Am. Math. Soc. 359(7), 3099–3123 (2007)
Shin, C., Sun, Q.: Stability of localized operator. J. Funct. Anal. 256(8), 2417–2439 (2009)
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)
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Communicated by Qiyu Sun.
Pengtong Li’s work was partially supported by National Natural Science Foundation of China (No. 10771101).
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Han, D., Li, P. & Tang, WS. Frames and their associated \(\emph{H}_{{\kern-2pt}\emph{F}}^{\emph{p}}\)-subspaces. Adv Comput Math 34, 185–200 (2011). https://doi.org/10.1007/s10444-010-9149-0
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DOI: https://doi.org/10.1007/s10444-010-9149-0