Embedding of non-polynomial spline spaces

Abstract

Abstract Purpose The aims of the paper are to obtain necessary and sufficient conditions ofexistence and smoothness for non-polynomial spline spaces of fifth order, toestablish the uniqueness of the B φ -spline spaces in the class C 4 among mentioned spaces (under condition of fixed grid), and toprove the embedding of the B φ -spline spaces corresponding to embedded grids. Methods In the paper, the approximation relations with initial grid and with completechain of vectors are applied to obtain the minimal spline spaces. Usage oflocally orthogonal chain of vectors gives opportunity to construct specialapproximation relations from which the initial space of B φ splines is constructed. Results Deletion of a knot from initial grid gives a new grid, and as result, a newspace of B φ splines is embedded in the initial space mentioned above. Conclusions Consequent deletion of the knots (one by one) generates the sequence of theembedded spaces of B φ splines. Obtained results are successfully proved. They may be appliedto spline-wavelet decompositions.