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Some Properties of Minimal Splines
Author(s) -
Demjanovich Yu. K.
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961770106
Subject(s) - mathematics , box spline , spline (mechanical) , equidistant , uniqueness , classification of discontinuities , hyperplane , grid , polynomial , arity , pure mathematics , mathematical analysis , spline interpolation , discrete mathematics , combinatorics , geometry , statistics , structural engineering , engineering , bilinear interpolation
S. G. Mikhlin was the first to construct systematically coordinate functions on an equidistant grid solving a system of approximate equations (called “fundamental relations”, see [5]; Goel discussed some special cases earlier in 1969; see also [1, 4, 6]). Further, the idea was developed in the case of irregular grids (which may have finite accumulation points, see [1] ). This paper is devoted to the investigation of A‐minimal splines, introduced by the author; they include polynomial minimal splines which have been discussed earlier. Using the idea mentioned above, we give necessary and sufficient conditions for existence, uniqueness and g‐continuity of these splines. The application of these results to polynomial splines of m ‐th degree on an equidistant grid leads us, in particular, to necessary and sufficient conditions for the continuity of their i ‐th derivative ( i = 1, ⃛, m ). These conditions do not exclude discontinuities of other derivatives (e.g. of order less than i ). This allows us to give a certain classification of minimal spline spaces. It turns out that the spline classes are in one‐to‐one‐correspondence with certain planes contained in a hyperplane.