Abstract
We prove a result about producing new frames for general spline-type spaces by piecing together portions of known frames. Using spline-type spaces as models for the range of certain integral transforms, we obtain results for time-frequency decompositions and sampling.
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Adams, R.A.: Sobolev Spaces. Pure Appl. Math., vol. 65. Academic Press, New York (1975)
Aldroubi, A., Cabrelli, C., Molter, U.M.: Wavelets on irregular grids with arbitrary dilation matrices, and frame atoms for L 2(R d). Appl. Comput. Harmon. Anal. 17, 119–140 (2004)
Balan, R.M., Casazza, P.G., Heil, C., Landau, Z.: Density, overcompleteness, and localization of frames I: Theory. J. Fourier Anal. Appl. 12(2), 105–143 (2006)
Baskakov, A.G.: Wiener’s theorem and the asymptotic estimates of the elements of inverse matrices. Funct. Anal. Appl. 24(3), 222–224 (1990)
Bownik, M.: The structure of shift-invariant subspaces of L 2(ℝn). J. Funct. Anal. 177(2), 282–309 (2000)
Casazza, P.G., Kutyniok, G.: Frames of subspaces. In: Wavelets, Frames and Operator Theory. Contemp. Math., vol. 345, pp. 87–113. Am. Math. Soc., Providence (2004)
Christensen, O., Eldar, Y.C.: Oblique dual frames and shift-invariant spaces. Appl. Comput. Harmon. Anal. 17(1), 48–68 (2004)
Christensen, O., Eldar, Y.C.: Generalized shift-invariant systems and frames for subspaces. J. Fourier Anal. Appl. 11(3), 299–313 (2005)
de Boor, C., DeVore, R.A., Ron, A.: The structure of finitely generated shift-invariant spaces in L 2(ℝd). J. Funct. Anal. 119(1), 37–78 (1994)
Dörfler, M.: Quilted frames—a new concept for adaptive representation, Preprint, ar**v:0912.2363 (2009)
Feichtinger, H.G.: Gewichtsfunktionen auf lokalkompakten Gruppen. Sitz. Österr. Akad. Wiss. 188, 451–471 (1979)
Feichtinger, H.G.: Banach convolution algebras of Wiener type. In: Sz.-Nagy, B., Szabados, J. (eds.) Proc. Conf. on Functions, Series, Operators, Budapest 1980. Colloq. Math. Soc. Janos Bolyai, vol. 35, pp. 509–524. North-Holland, Amsterdam (1983)
Feichtinger, H.G.: Banach spaces of distributions defined by decomposition methods, II. Math. Nachr. 132, 207–237 (1987)
Feichtinger, H.G.: Wiener amalgams over Euclidean spaces and some of their applications. Lect. Notes Pure Appl. Math. 136, 123–137 (1992)
Feichtinger, H.G.: Spline-type spaces in Gabor analysis. In: Zhou, D.X. (ed.) Wavelet Analysis: Twenty Years Developments Proceedings of the International Conference of Computational Harmonic Analysis, Hong Kong, China, June 4–8, 2001. Ser. Anal., vol. 1, pp. 100–122. World Scientific, River Edge (2002)
Feichtinger, H.G., Gröbner, P.: Banach spaces of distributions defined by decomposition methods, I. Math. Nachr. 123, 97–120 (1985)
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86, 307–340 (1989)
Feichtinger, H.G., Strohmer, T.: Advances in Gabor Analysis. Birkhäuser, Basel (2003)
Feichtinger, H.G., Zimmermann, G.: A Banach space of test functions for Gabor analysis. In: Feichtinger, H., Strohmer, T. (eds.) Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, pp. 123–170. Birkhäuser, Boston (1998)
Fornasier, M.: Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl. 289(1), 180–199 (2004)
Fornasier, M., Gröchenig, K.: Intrinsic localization of frames. Constr. Approx. 22(3), 395–415 (2005)
Frazier, M.W., Jawerth, B.D., Weiss, G.: Littlewood-Paley Theory and the Study of Function Spaces. Am. Math. Soc., Providence (1991)
Grafakos, L.: Classical and Modern Fourier Analysis. Prentice Hall, New York (2004)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston (2001)
Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004)
Gröchenig, K., Fendler, G., Leinert, M.: Convolution-dominated operators on discrete groups. Integr. Equ. Oper. Theory 61, 493–509 (2008)
Gröchenig, K., Leinert, M.: Symmetry and inverse-closedness of matrix algebras and symbolic calculus for infinite matrices. Trans. Am. Math. Soc. 358, 2695–2711 (2006)
Gröchenig, K., Piotrowski, M.: Molecules in coorbit spaces and boundedness of operators. Studia Math. 192(1), 61–77 (2009)
Jaffard, S.: Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. (Properties of matrices “well localized” near the diagonal and some applications). Ann. Inst. Henri Poincaré Anal. Non Linéaire 7(5), 461–476 (1990)
Oswald, P.: Frames and space splittings in Hilbert spaces. Technical Report (1997)
Romero, J.L.: Explicit localization estimates for spline-type spaces. Sampl. Theory Signal Image Process. 8(3), 249–259 (2009)
Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces of L 2(ℝd). Can. J. Math. 47(5), 1051–1094 (1995)
Ron, A., Shen, Z.: Affine systems in L 2(ℝd): the analysis of the analysis operator. J. Funct. Anal. 148(2), 408–447 (1997)
Sun, Q.: Wiener’s lemma for infinite matrices with polynomial off-diagonal decay. C. R. Math. Acad. Sci. Paris 340(8), 567–570 (2005)
Sun, Q.: Wiener’s lemma for infinite matrices. Trans. Am. Math. Soc. 359(7), 3099 (2007)
Sun, Q.: Frames in spaces with finite rate of innovation. Adv. Comput. Math. 28(4), 301–329 (2008)
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Communicated by Peter G. Casazza.
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Romero, J.L. Surgery of Spline-type and Molecular Frames. J Fourier Anal Appl 17, 135–174 (2011). https://doi.org/10.1007/s00041-010-9127-4
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DOI: https://doi.org/10.1007/s00041-010-9127-4