Analytic functions in local shift-invariant spaces and analytic limits of level dependent subdivision. (English) Zbl 1476.46037

Charina, Maria; Protasov, Vladimir Yu.

J. Fourier Anal. Appl. 27, No. 3, Paper No. 45, 31 p. (2021).

Summary: In this paper we characterize all subspaces of analytic functions in finitely generated shift-invariant spaces with compactly supported generators and provide explicit descriptions of their elements. We illustrate the differences between our characterizations and Strang-Fix or zero conditions on several examples. Consequently, we depict the analytic functions generated by scalar or vector subdivision with masks of bounded and unbounded support. In particular, we prove that exponential polynomials are indeed the only analytic limits of level dependent scalar subdivision schemes with finitely supported masks.

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
41A30	Approximation by other special function classes
42B35	Function spaces arising in harmonic analysis
65D17	Computer-aided design (modeling of curves and surfaces)

Keywords:

analytic subspaces; finitely generated shift-invariant spaces; refinable functions; level dependent (non-stationary)

References:

[1]	Aldroubi, A.; Gröchenig, K., Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43, 585-620 (2001) · Zbl 0995.42022
[2]	Bendory, T.; Dekel, S.; Feuer, A., Exact recovery of non-uniform splines from the projection onto spaces of algebraic polynomials, J. Approx. Theory, 182, 7-17 (2014) · Zbl 1290.41002
[3]	de Boor, C.; DeVore, RA; Ron, A., Approximation from shift-invariant subspaces of \(L_2({\mathbb{R}}^d)\), Trans. Am. Math. Soc., 341, 787-806 (1994) · Zbl 0790.41012
[4]	de Boor, C.; DeVore, RA; Ron, A., The structure of finitely generated shift-invariant spaces in \(L_2({\mathbb{R}}^d)\), J. Funct. Anal., 119, 37-78 (1994) · Zbl 0806.46030
[5]	de Boor, C.; DeVore, RA; Ron, A., Approximation orders of FSI spaces in \(L_2({\mathbb{R}}^d)\), Constr. Approx., 14, 631-652 (1998) · Zbl 0919.41009
[6]	Cabrelli, C.; Heil, C.; Molter, U., Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory, 95, 5-52 (1998) · Zbl 0911.41008
[7]	Cabrelli, CA; Heil, C.; Molter, UM, Self-similarity and multiwavelets in higher dimensions, Mem. Am. Math. Soc., 170, 807 (2004) · Zbl 1063.42024
[8]	Cavaretta, AS; Dahmen, W.; Micchelli, CA, Stationary Subdivision, Mem. Am. Math. Soc., 453, 1-185 (1991) · Zbl 0741.41009
[9]	Charina, M.; Conti, C.; Guglielmi, N.; Protasov, VY, Regularity of non-stationary subdivision: a matrix approach, Numer. Math., 135, 639-678 (2017) · Zbl 06695813
[10]	Charina, M.; Conti, C.; Jetter, K.; Zimmermann, G., Scalar multivariate subdivision schemes and Box splines, J. CAGD, 28, 285-306 (2011) · Zbl 1221.65060
[11]	Charina, M.; Conti, C.; Romani, L., Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix, Numer. Math., 127, 223-254 (2014) · Zbl 1295.65018
[12]	Chui, CK; He, W.; Stöckler, J., Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comp. Harmonic Anal., 13, 224-262 (2002) · Zbl 1016.42023
[13]	Chui, C.; de Viles, J., Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces (2011), Boca Raton: CRC Press, Boca Raton · Zbl 1202.42001
[14]	Cohen, A.; Dyn, N., Nonstationary subdivision schemes and multiresolution analysis, SIAM J. Math. Anal., 27, 1745-1769 (1996) · Zbl 0862.41013
[15]	Colella, D.; Heil, C., Characterizations of scaling functions: continuous solutions, SIAM J. Matrix Anal. Appl., 15, 469-518 (1994) · Zbl 0797.39006
[16]	Conti, C.; Cotronei, M.; Romani, L., Beyond B-splines: exponential pseudo-splines and subdivision schemes reproducing exponential polynomials, Dolomities Res. Notes Approx., 10, 31-42 (2017) · Zbl 1370.41019
[17]	Conti, C.; Romani, L., Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction, J. Comput. Appl. Math., 236, 543-556 (2011) · Zbl 1232.65035
[18]	Conti, C.; Romani, L.; Yoon, J., Approximation order and approximate sum rules in subdivision, J. Approx. Theory, 207, 380-401 (2016) · Zbl 1338.65045
[19]	Conway, JB, Functions of one Complex Variable (1978), Berlin: Springer, Berlin
[20]	Dai, XR; Song, JQ, Summation and intersection of refinable shift invariant spaces, Sci. China Math., 54, 20-87 (2011) · Zbl 1247.42038
[21]	Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics (1992), Philadelphia: SIAM, Philadelphia · Zbl 0776.42018
[22]	Daubechies, I.; Han, B.; Ron, A.; Shen, Z., Framelets: MRA-based constructions of wavelet frames, Appl. Comp. Harmonic Anal., 14, 1-46 (2003) · Zbl 1035.42031
[23]	Dyn, N.; Light, WA, Subdivision schemes in CAGD, Advances in Numerical Analysis Vol. II: Wavelets, Subdivision Algorithms and Radial Basis Functions, 36-104 (1992), Oxford: Oxford University Press, Oxford
[24]	Dyn, N.; Levin, D., Subdivision schemes in geometric modeling, Acta Numer., 11, 73-144 (2002) · Zbl 1105.65310
[25]	Dyn, N.; Levin, D.; Luzzatto, A., Exponentials reproducing subdivision schemes, Found. Comput. Math., 3, 187-206 (2003) · Zbl 1095.41001
[26]	Dyn, N.; Ron, A., Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems, Trans. Am. Math. Soc., 319, 381-403 (1990) · Zbl 0699.41012
[27]	Filippov, VI; Oswald, P., Representation in \(L_p\) by series of translates and dilates of one function, J. Approx. Theory, 82, 15-29 (1995) · Zbl 0861.42022
[28]	Han, B., Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory, 124, 44-88 (2003) · Zbl 1028.42019
[29]	Han, B.; Shen, Z., Compactly supported symmetric \(C^\infty\) wavelets with spectral approximation order, SIAM J. Math. Anal., 40, 905-938 (2008) · Zbl 1161.42312
[30]	Han, B.; Shen, Z., Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames, Isr. J. Math., 172, 371-398 (2009) · Zbl 1180.42004
[31]	Han, B., Framelets and wavelets: Algorithms, analysis, and applications. Applied and Numerical Harmonic Analysis (2017), Cham: Birkhäuser/Springer, Cham · Zbl 1387.42001
[32]	Han, B.; Jia, R-Q, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29, 1177-1199 (1998) · Zbl 0915.65143
[33]	Han, B., Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory, 110, 18-53 (2001) · Zbl 0986.42020
[34]	Jetter, K.; Plonka, G.; Dyn, N.; Leviatan, D.; Levin, D.; Pinkus, A., A survey on \(L_2\)-approximation orders from shift-invariant spaces, Multivariate Approximation and Applications, 73-111 (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 1039.42033
[35]	Jia, R-Q, Subdivision schemes in \(L_p\) spaces, Adv. Comput. Math., 3, 309-341 (1995) · Zbl 0833.65148
[36]	Jia, RQ, Shift-invariant spaces on the real line, Proc. Am. Math. Soc, 125, 785-793 (1997) · Zbl 0860.41021
[37]	Jia, RQ, Approximation properties of multivariate wavelets, Math. Comp., 67, 647-665 (1998) · Zbl 0889.41013
[38]	Jia, RQ; Jiang, Q., Approximation power of refinable vectors of functions, Wavelet Analysis and Applications, Studies in Advanced Mathematics 25, 155-178 (2002), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0997.42022
[39]	Levin, A., Polynomial generation and quasi-interpolation in stationary non-uniform subdivision, Comput. Aided Geom. Des., 20, 41-60 (2003) · Zbl 1069.41503
[40]	Micchelli, CA; Sauer, T., Regularity of multiwavelets, Adv. Comp. Math., 7, 455-545 (1997) · Zbl 0902.65095
[41]	Peter, J.; Reif, U., Subdivision Surfaces. Geometry and Computing (2008), Berlin: Springer-Verlag, Berlin
[42]	Plonka, G., Approximation order provided by refinable function vectors, Constr. Approx., 13, 221-224 (1997) · Zbl 0870.41015
[43]	Plonka, G.; Ron, A., A new factorization technique of the matrix mask of univariate refinable functions, Numer. Math., 87, 555-595 (2001) · Zbl 0982.65150
[44]	Novikov, I.Y., Protasov, V.Y., Skorpina, M.A.: Wavelet Theory. AMS, Translations of mathematical monographs, p. 239 (2011)
[45]	Protasov, VY, On the decay of infinite products of trigonometric polynomials, Math. Notes, 72, 819-832 (2002) · Zbl 1023.42002
[46]	Ron, A., Exponential box splines, Constr. Approx., 4, 357-378 (1988) · Zbl 0674.41005
[47]	Ron, A.; Pinkus, A.; Leviatan, D.; Dyn, N.; Levin, D., Introduction to shift-invariant spaces: Linear independence, Multivariate Approximation and Applications, 112-151 (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 1041.42031
[48]	Ron, A.; Shen, Z., Affine systems in \(L_2({\mathbb{R}}^d)\): the analysis of the analysis operator, J. Func. Anal., 148, 408-447 (1997) · Zbl 0891.42018
[49]	Ron, A.; Shen, Z., Frames and stable bases for shift invariant subspaces of \(L_2({\mathbb{R}}^d)\), Can. J. Math., 47, 1051-1094 (1995) · Zbl 0838.42016
[50]	Rvachev, VA, Compactly supported solutions of functional-differential equations and their applications, Russ. Math. Surv., 45, 87-120 (1990) · Zbl 0704.34090
[51]	Schoenberg, IJ, Contributions to the problem of approximation of equidistant data by analytic functions, Parts A and B, Quart. Appl. Math., 4, 45-99, 112-141 (1946)
[52]	Shen, Z., Refinable function vectors, SIAM J. Math. Anal., 29, 235-250 (1998) · Zbl 0913.42028
[53]	Strang, G., Fix, G.: A Fourier analysis of the finite element variational method, C.I.M.E. II, Ciclo 1971, Constructive Aspects of Functional Analysis, pp. 793-840 (1973) · Zbl 0272.65099
[54]	Stoica, P.; Moses, RL, Spectral Analysis of Signals (2005), Upper Saddle River, NJ: Pearson/Prentice Hall, Upper Saddle River, NJ
[55]	Terekhin, PA, Affine synthesis in the space \(L_2(R^d)\), Izv. Math., 73, 171-180 (2009) · Zbl 1175.41010
[56]	Vetterli, M.; Marziliano, P.; Blu, T., Sampling signals with finite rate of innovation, IEEE Trans. Signal Process, 50, 1417-1428 (2002) · Zbl 1369.94309
[57]	Vonesch, C.; Blu, T.; Unser, M., Generalized Daubechies wavelet families, IEEE Trans. Signal Process., 55, 4415-4429 (2007) · Zbl 1390.42051
[58]	Warren, J.; Weimer, H., Subdivision Methods for Geometric Design (2002), Burlington: Morgan-Kaufmann, Burlington