Exponential box splines

We consider certain exponential box splines E on a three-direction mesh whose exponents satisfy a symmetry condition. It is shown, in particular, that given bounded data on the integer lattice in R2, there is a unique bounded combination of integer translates of E that interpolates the data. When all exponents are zero, this reduces to a result of de Boor, Höllig and Riemenschneider in [2]. Unlike the proof in [2] we use only elementary analysis and do not employ any computer calculations.

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Translates of exponential box splines and their related spaces

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1988-0961608-7 ◽

1988 ◽

Vol 309 (2) ◽

pp. 683-683 ◽

Cited By ~ 22

Author(s):

Asher Ben-Artzi ◽

Amos Ron

Keyword(s):

Box Splines ◽

Exponential Box Splines

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Concerning the linear dependence of integer translates of exponential box splines

Journal of Approximation Theory ◽

10.1016/0021-9045(91)90088-r ◽

1991 ◽

Vol 64 (1) ◽

pp. 95-118 ◽

Cited By ~ 3

Author(s):

N Sivakumar

Keyword(s):

Linear Dependence ◽

Box Splines ◽

Integer Translates ◽

Exponential Box Splines

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Homogeneous polynomial splines

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027621 ◽

1991 ◽

Vol 117 (1-2) ◽

pp. 89-102 ◽

Cited By ~ 2

Author(s):

T. N. T. Goodman ◽

S. L. Lee

Keyword(s):

Three Dimensions ◽

Piecewise Polynomial ◽

Polynomial Splines ◽

Homogeneous Polynomials ◽

Box Splines ◽

Functions Of Two Variables ◽

Piecewise Polynomial Functions ◽

Exponential Box Splines ◽

Limiting Case ◽

Polynomial Box Splines

SynopsisWe construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions. These give a rich and elegant theory which combines properties of polynomial box splines see [6] and the references therein) with the explicit representation of simple exponential box splines [11], while enjoying complete symmetry in the three variables. By a linear transformation followed by a projection on suitable planes, one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines. The vertices of these pencils may be finite or one or two may be infinite, i.e. the corresponding pencils may comprise parallel lines. As a limiting case, all three vertices become infinite and one recovers polynomial box splines on a three-direction mesh.

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