The aim of this paper is to calculate relative and absolute projection constants of certain subspaces of and generated by eigenvectors of sign matrices. The main tool in our considerations is so called Chalmers–Metcalf operator (see Chalmers & Metcalf (1994) and Lewicki & Prophet (2021)). Also, some results from Castejon & Lewicki (2014) and Castejon & Lewicki (2019) will be applied.
We consider best approximation by rational functions in Musielak–Orlicz spaces of real-valued measurable functions over the unit interval equipped with the Lebesgue measure. We prove several properties of the respective multi-value projection operator, including its semi-continuity. Our results generalise known results for Lebesgue and variable Lebesgues spaces, and can be applied to special cases including Orlicz spaces and variable Lebesgue spaces with weights. We touch upon applications to image processing.
The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of as the closure of in different senses and the extension of continuous vector-valued functions are studied.
Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.
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