TY - JOUR

T1 - On bivariate smoothness spaces associated with nonlinear approximation

AU - Dekel, S.

AU - Leviatan, D.

AU - Sharir, M.

PY - 2004

Y1 - 2004

N2 - In recent years there have been various attempts at the representations of v multivariate signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities and applying bandlets, curvelets, ridgelets, etc. (e.g., [3], [4], [8], [15], [18], [26], [27], [29]), or allocating some weights to curves of singularities like the Mumford-Shah functional [25] and its modifications. In the latter approach a function is approximated on subdomains where it is smoother but there is a penalty in the form of the total length (or other measurement) of the partitioning curves. We introduce a combined measure of smoothness of the function in several dimensions by augmenting its smoothness on subdomains by the smoothness of the partitioning curves. Also, it is known that classical smoothness spaces fail to characterize approximation spaces corresponding to multivariate piecewise polynomial nonlinear approximation. We show how the proposed notion of smoothness can almost characterize these spaces. The question whether the characterization proposed in this work can be further "simplified" remains open.

AB - In recent years there have been various attempts at the representations of v multivariate signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities and applying bandlets, curvelets, ridgelets, etc. (e.g., [3], [4], [8], [15], [18], [26], [27], [29]), or allocating some weights to curves of singularities like the Mumford-Shah functional [25] and its modifications. In the latter approach a function is approximated on subdomains where it is smoother but there is a penalty in the form of the total length (or other measurement) of the partitioning curves. We introduce a combined measure of smoothness of the function in several dimensions by augmenting its smoothness on subdomains by the smoothness of the partitioning curves. Also, it is known that classical smoothness spaces fail to characterize approximation spaces corresponding to multivariate piecewise polynomial nonlinear approximation. We show how the proposed notion of smoothness can almost characterize these spaces. The question whether the characterization proposed in this work can be further "simplified" remains open.

KW - Besov spaces

KW - K-Functional

KW - Modulus of smoothness

KW - Multivariate nonlinear approximation

KW - Mumford-Shah functional

KW - Piecewise polynomials approximation

KW - Smoothness spaces

KW - Wavelets

UR - http://www.scopus.com/inward/record.url?scp=4644353369&partnerID=8YFLogxK

U2 - 10.1007/s00365-003-0549-y

DO - 10.1007/s00365-003-0549-y

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AN - SCOPUS:4644353369

SN - 0176-4276

VL - 20

SP - 625

EP - 646

JO - Constructive Approximation

JF - Constructive Approximation

IS - 4

ER -