Wavelet decompositions on nonuniform grids

Embedding and wavelet decomposition of spaces of minimal splines

“…In this paper, we use the notation, terminology, and some results of [12,13]. For the sake of convenience, we formulate some results used below.…”

“…We omit the proof of formulas (6.14)-(6.22) since the proof of Theorems 6 and 7 is similar to that of Theorems 4 and 5 in [13]. Corollary 6.…”

“…The required assertion is easily obtained by applying the functional g (k) to the approximation relations (2.2) (cf. [13]).…”

“…In this case, the corresponding wavelet decomposition was also obtained (cf. [13]). We note that the chains of embedded spaces and their wavelet decompositions were obtained by the author [11] in the case of uniform grids and minimal polynomial splines (not necessarily continuous).…”

Calibration relations for B φ -splines of fourth order

“…The central structure of wavelet construction is a chain of embedded spaces. The construction of such chains for the space of (nonpolynomial) minimal splines on nonuniform grids was studied in [9]- [12], where such chains were constructed for smooth spaces of splines of first, second, and third order. Spaces of splines of any order (including those of smooth splines) were constructed in [12], where an embedding was established only for special (nonsmooth) spaces of splines.…”