Biorthogonal spline-wavelets on the interval - stability and moment conditions

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0, 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on R developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly stable bases on each level. As a second step these initial complements are then projected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal as well as on the dual side have finitely supported masks the corresponding decomposition and reconstruction algorithms are simple and efficient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases for L_2 ([0, 1]) and that weighted sequence norms for the coefficients of such wavelet expansions characterize Sobolev spaces and their duals on [0, 1] within a range depending on the parameters in the Jackson and Bernstein estimates. (orig.)Available from TIB Hannover: RR 5549(265)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Blended Learning and Sustainable Development

The concept of blended learning has its origins in the 1960’s when new options for technology-mediated education became available to complement conventional approaches to face-to-face teaching in higher education and other contexts. In HE institutions, blended learning has now become so widespread that many regard it as the “new normal” in describing HE approaches to teaching and learning. However, the ubiquity of the blended learning concept in HE is problematic for establishing a common understanding of the concept, which is essential for identifying how it contributes to sustainable development. Blended learning may be defined as the design of learning experiences that draw on a combination of face-face, distance, or online delivery methods, learning technologies, delivery multimedia, and pedagogical methodologies to achieve a mix of learning outcomes in educational or training contexts. Blended learning designs can support sustainable development, including the social, economic, and environmental dimensions of sustainability and protect global environmental resources to meet the needs of the present and future generations. This chapter first examines the various conceptualisations of blended learning in HE teaching and learning systems, and then proceeds to examine contemporary understanding of how HE blended learning provision contributes or might contribute to sustainable development and to the Sustainable Development Goals (SDGs)