We study necessary and sufficient conditions for the smoothness of (not necessarily polynomial) splines of fourth order. We establish the uniqueness of a space of splines of maximal smoothness and prove embeddings on refined grids. We also obtain the corresponding calibration relations. Bibliography: 12 titles.
We consider different methods for constructing a chain of embedded spaces of first order splines (not necessarily smooth and polynomial), based on a local enlargement of an irregular grid. We present the corresponding wavelet decompositions and prove that the decomposition operators are commutative relative to the order of removing grid points. Bibliography: 6 titles.Embedding of spaces of smooth polynomial splines and the corresponding wavelet decompositions on infinite embedded grids were studied in many works (cf., for example, [1]-[3] and the references therein). In particular, smooth nonpolynomial splines were considered in [4]- [6]. A grid can be enlarged by removing grid points one at a time. Hence the question arises, whether the corresponding decomposition operation is commutative. Such considerations are based on approximate relations, owing to which it is possible to obtain wavelet decompositions of spline spaces with different smoothness and such that their approximate properties are asymptotically optimal with respect to the N -diameter of standard compact sets. The original numerical flow is regarded as a sequence of coefficients of decomposition with respect to the coordinate splines in the space constructed on the original (small) grid. This space is projected onto the embedded spline space (on an enlarged grid). As a result, we get a grid obtained by splitting the original numerical information flow into the basic flow (formed by the coefficients of decomposition relative to the coordinate splines of the embedded space) and the wavelet numerical flow which can be used to restore the original numerical flow. Nonsmooth splines are simpler than smooth ones; wavelet decompositions of a space of such splines on a segment (with a finite grid) are good for computations purposes. In this paper, based on a local enlargement of an irregular grid, we present a simple method for constructing chains of embedded spline spaces on a segment.
We consider conditions for embedding of (generally speaking, discontinuous and nonpolynomial) spline spaces obtained by a removal of grid points (nests). For such spaces we present the wavelet decomposition, construct the embedding and extension matrices, and derive the corresponding decomposition and reconstruction formulas. We also consider the decomposition and restoration operators in spaces of finite sequences (flows). Bibliography: 1 title.In [1], we considered nonsmooth first order spline-wavelet decompositions obtained by a successive removal of grid points. This allowed us to simplify (comparing with smooth decompositions) formulas for computing these decompositions in such a way that the adaptivity properties of the handled flows are preserved. As was shown In [1], the wavelet decomposition is independent of the order of removing grid points. However, if we realize the process numerically, in the machine arithmetic with floating decimal point, a successive removal of a large number of grid points leads to a fast buildup of rounding errors. Hence it is reasonable to apply such an approach only for computing in the real time scale, when a delay is not admissible.In this paper, we consider the situation where delays in the processing of incoming numerical flows are admitted. In this case, it is actual to remove simultaneously groups of grid points, called nests. We consider the embedding conditions for (in general, discontinuous and nonpolynomial) spline spaces obtained by eliminating a group of grid points (nests), present their wavelet decomposition, and construct the embedding and extension matrices, as well as the corresponding decomposition and reconstruction formulas. We also consider the decomposition and restoration operators in spaces of finite sequences (flows). We deal with a single nest, but, in the last section, we show how to extend the obtained results to a set of nests.
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