TY - JOUR

T1 - Some properties of minimal splines

AU - Demjanovich, Yu K.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0039021350&partnerID=8YFLogxK

U2 - 10.1002/mana.19961770106

DO - 10.1002/mana.19961770106

M3 - Article

AN - SCOPUS:0039021350

VL - 177

SP - 57

EP - 79

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

ER -