An existence result on Ky Fan type best approximation is proved. For this purpose, a class of factorizable multifunctions and the other one being a demicontinuous, relative almost quasi-convex, onto function on an approximately weakly compact, convex subset of Hausdorff locally convex topological vector space are used. As consequence, this result extends the best approximation results of Basha and Veeramani^[8] and many others.

The main objective of this work is to decompose orthogonally the reproducing kernels Hilbert space using any conditionally positive definite kernels into smaller ones by introducing the theory of power kernels, and to show how to do this decomposition recursively. It may be used to split large interpolation problems into smaller ones with different kernels which are related to the original kernels. To reach this objective, we will reconstruct the reproducing kernels Hilbert space for the normalized and the extended kernels and give the recursive algorithm of this decomposition.

We consider the problem $K(x)u_{xx} = u_{tt}$, $0 < x < 1$, $t \geq 0$, with the boundary condition $u(0, t) = g(t) \in L^2(R)$ and $u_x(0, t) = 0$, where $K(x)$ is continuous and $0 < \alpha \leq K(x) < +\infty$. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on $g$. Considering the existence of a solution $u(x, \cdot) \in H^2(R)$ and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore, we prove the uniqueness of the solution for this problem.

In the present paper, we obtain estimations of convergence rate derivatives of the $q$-Bernstein polynomials $B_n(f,q_n;x)$ approximating to $f'(x)$ as $n\to\infty$ which is a generalization of that relating the classical case $q_n = 1$. On the other hand, we study the convergence properties of derivatives of the limit $q$-Bernstein operators $B_\infty( f,q;x)$ as $q\to 1^−.$

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $\lambda\to T \to \lambda S$ are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.

Let $p(z)$ be a polynomial of degree at most $n$. In this paper we obtain some new results about the dependence of$$\Bigg\|p(Rz)−\beta p(rz)+\alpha\Big\{\frac{R+1}{r+1}\Big)^n-|\beta|\Big\} p(rz)\Bigg\|_s$$ on $\|p(z)\|_s$ for every $\alpha$, $\beta \in C$ with $|\alpha| \leq 1$, $|\beta| \leq 1$, $R > r \ge 1$, and $s > 0$. Our results not only generalize some well known inequalities, but also are variety of interesting results deduced from them by a fairly uniform procedure.

In the present paper we consider quartic piecewise polynomial for approximation to the function $f ∈ C^2[0,1]$. A convex type condition has been imposed in the partition so that the matrix involved for the computation of pp functions is of lower band. This reduces the computation for constructions of the pp functions for the approximation.

In this paper we prove some interesting extensions and generalizations of Enestrom-Kakeya Theorem concerning the location of the zeros of a polynomial in a complex plane. We also obtain some zero-free regions for a class of related analytic functions. Our results not only contain some known results as a special case but also a variety of interesting results can be deduced in a unified way by various choices of the parameters.

In this paper, we investigate $n$-dimensional complete and orientable hypersurfaces $M^n$ $(n \geq3)$ with constant normalized scalar curvature in a locally symmetric manifold. Two rigidity theorems are obtained for these hypersurfaces.