## Approximation orders of shift-invariant subspaces of $$W_{2}^{s}(\mathbb R^d)$$.(English)Zbl 1092.41010

The authors discuss approximation orders of shift-invariant spaces in Sobolev spaces. First, they characterize in Theorem 21 the approximation orders provided by finitely generated shift-invariant (FSI) spaces by means of the Fourier transforms of the generators, extending a result of de C. Boor, R. A. de Vore and A. Ron [Trans. Am. Math. Soc. 341, No. 2, 787–806 (1994; Zbl 0790.41012)] from the $$L^2$$ space to Sobolev spaces. Then they prove the existence of a superfunction and discuss its behavior (good or bad superfunction). Finally, approximation orders, Strang-Fix type conditions, and polynomial reproductions are investigated for FSI generators associated with matrix refinement equations.

### MSC:

 41A25 Rate of convergence, degree of approximation 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Zbl 0790.41012
Full Text:

### References:

 [1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 [2] de Boor, C., The polynomials in the linear span of integer translates of a compactly supported function, Constr. approx., 3, 2, 199-208, (1987) · Zbl 0624.41013 [3] de Boor, C.; DeVore, R.A.; Ron, A., Approximation from shift-invariant sub-spaces of $$L_2(\mathbb{R}^d)$$, Trans. amer. math. soc., 341, 2, 787-806, (1994) · Zbl 0790.41012 [4] de Boor, C.; DeVore, R.A.; Ron, A., The structure of finitely generated shift-invariant subspaces in $$L_2(\mathbb{R}^d)$$, J. funct. anal., 119, 1, 37-78, (1994) · Zbl 0806.46030 [5] de Boor, C.; DeVore, R.A.; Ron, A., Approximation orders of FSI spaces in $$L_2(\mathbb{R}^d)$$, Constr. approx., 14, 4, 631-652, (1998) · Zbl 0919.41009 [6] de Boor, C.; Höllig, K., Approximation order from bivariate $$C^1$$-cubicsa counterexample, Proc. amer. math. soc., 87, 4, 649-655, (1983) · Zbl 0545.41017 [7] de Boor, C.; Ron, A., The exponentials in the span of the multiinteger translates of a compactly supported functionquasiinterpolation and approximation order, J. London math. soc. (2), 45, 3, 519-535, (1992) · Zbl 0757.41012 [8] de Boor, C.; Ron, A., Fourier analysis of the approximation power of principal shift-invariant spaces, Constr. approx., 8, 4, 427-462, (1992) · Zbl 0801.41027 [9] Cabrelli, C.A.; Heil, C.; Molter, U., Polynomial reproduction by refinable functions, (), 121-161 [10] Cabrelli, C.A.; Heil, C.; Molter, U., Accuracy of lattice translates of several multidimensional refinable functions, J. approx. theory, 95, 1, 5-52, (1998) · Zbl 0911.41008 [11] Cabrelli, C.; Heil, C.; Molter, U., Accuracy of several multidimensional refinable distributions, J. Fourier anal. appl., 6, 5, 483-502, (2000) · Zbl 0960.42016 [12] Cavaretta, A.S.; Dahmen, W.; Micchelli, C.A., Stationary subdivision, () · Zbl 0741.41009 [13] Chui, C.K.; Jetter, K.; Ward, J.D., Cardinal interpolation by multivariate splines, Math. comp., 48, 711-724, (1987) · Zbl 0619.41004 [14] Heil, C.; Strang, G.; Strela, V., Approximation by translates of refinable functions, Numer. math., 73, 1, 75-94, (1996) · Zbl 0857.65015 [15] Jetter, K.; Plonka, G., A survey of $$L_2$$-approximation order from shift-invariant spaces, (), 73-111 · Zbl 1039.42033 [16] Jetter, K.; Zhou, D.-X., Seminorm and full norm order of linear approximation from shift-invariant spaces, Rend. sem. mat. fis. milano, 65, 277-302, (1995) · Zbl 0882.41025 [17] Ji, H.; Riemenschneider, S.; Shen, Z., Multivariate compactly supported fundamental refinable functions, duals, and biorthogonal wavelets, Stud. appl. math., 102, 2, 173-204, (1999) · Zbl 1005.42019 [18] Jia, R.-Q., Shift-invariant spaces and linear operator equations, Israel J. math., 103, 259-288, (1998) · Zbl 0927.41011 [19] R.-Q. Jia, Refinable shift-invariant spaces: from splines to wavelets, in: C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory VIII, vol. 2 (College Station, TX, 1995), Ser. Approx. Decompos., vol. 6, World Scientific Publishing, River Edge, NJ, 1995, pp. 179-208. · Zbl 0927.42021 [20] Jia, R.-Q., Approximation properties of multivariate wavelets, Math. comp., 67, 222, 647-665, (1998) · Zbl 0889.41013 [21] Jia, R.-Q.; Jiang, Q.-T.; Shen, Z., Distributional solutions of nonhomogeneous discrete and continuous refinement equations, SIAM J. math. anal., 32, 2, 420-434, (2000) · Zbl 0980.42035 [22] Jia, R.Q.; Micchelli, C.A., Using the refinement equations for the construction of pre-wavelets. II. powers of two, (), 209-246 · Zbl 0777.41013 [23] Jia, R.-Q.; Riemenschneider, S.D.; Zhou, D.-X., Approximation by multiple refinable functions, Canad. J. math., 49, 5, 944-962, (1997) · Zbl 0904.41010 [24] Jiang, Q.-T.; Shen, Z., On existence and weak stability of matrix refinable functions, Constr. approx., 15, 3, 337-353, (1999) · Zbl 0932.42028 [25] Johnson, M.J., On the approximation order of principal shift-invariant subspaces of $$L_p(\mathbb{R}^d)$$, J. approx. theory, 91, 3, 279-319, (1997) · Zbl 0891.41008 [26] Johnson, M.J., An upper bound on the approximation power of principal shift-invariant spaces, Constr. approx., 13, 2, 155-176, (1997) · Zbl 0876.31004 [27] Kyriazis, G.C., Wavelet-type decompositions and approximations from shift-invariant spaces, J. approx. theory, 88, 2, 257-271, (1997) · Zbl 0878.42016 [28] Y. Meyer, Ondelettes et Opérateurs I: Ondelettes, Hermann Éditeurs, 1990. [29] Micchelli, C.A., Using the refinement equation for the construction of pre-wavelets, Numer. algorithms, 1, 1, 75-116, (1991) · Zbl 0759.65005 [30] Plonka, G., Approximation order provided by refinable function vectors, Constr. approx., 13, 2, 221-244, (1997) · Zbl 0870.41015 [31] Ramanathan, J., Methods of applied Fourier analysis, (1998), Birkhäuser Boston · Zbl 0926.94004 [32] Riemenschneider, S.; Shen, Zuowei, Wavelets and pre-wavelets in low dimensions, J. approx. theory, 71, 1, 18-38, (1992) · Zbl 0772.41019 [33] Ron, A., Smooth refinable functions provide good approximation orders, SIAM J. math. anal., 28, 3, 731-748, (1997) · Zbl 0876.42028 [34] Ron, A.; Shen, Z., Frames and stable bases for shift-invariant subspaces of $$L_2(\mathbb{R}^d)$$, Canad. J. math., 47, 5, 1051-1094, (1995) · Zbl 0838.42016 [35] Ron, A.; Sivakumar, N., The approximation order of box spline spaces, Proc. amer. math. soc., 117, 2, 473-482, (1993) · Zbl 0766.41012 [36] Rudin, W., Real and complex analysis, (1987), WCB/McGraw-Hill Boston · Zbl 0925.00005 [37] Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions, parts A & B, Quart. appl. math., IV, 45-99, 112-141, (1946) [38] G. Strang, G. Fix, A Fourier analysis of the finite-element variational method, in: Constructive Aspects of Functional Analysis, C.I.M.E., 1973, pp. 793-840. · Zbl 0278.65116 [39] Zhao, K., Simultaneous approximation from PSI spaces, J. approx. theory, 81, 2, 166-184, (1995) · Zbl 0821.41019 [40] Zhao, K., Simultaneous approximation and quasi-interpolants, J. approx. theory, 85, 2, 201-217, (1996) · Zbl 0920.41008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.