We study the problem of best local polynomial approximation in variable Lebesgue spaces L^p(· ), where the error is measured by an averaged Luxemburg seminorm over measurable sets shrinking to a finite collection of prescribed nodes. In this context, we establish a Pólya-type inequality for algebraic polynomials and develop uniform boundedness and error estimates for nets of best approximants. These results provide the basis for analyzing the asymptotic behavior of best local approximations and their connection to Hermite interpolation. When the number of prescribed nodes does not exceed the dimension of the polynomial space, we identify the limiting approximation explicitly. In the overdetermined case, we characterize the cluster points as solutions of finite-dimensional minimization problems. As an application, we obtain sufficient conditions for the existence of best local approximations of piecewise smooth functions in variable Lebesgue spaces.
The study of linear combinations of weighted composition operators on various analytic function spaces is a natural, yet challenging topic in operator theory. Motivated by the work of Choe, Koo and Wang, we establish estimates for the essential norm of linear combinations of weighted composition operators acting on weighted Besov–Sobolev spaces. Our method integrates reproducing kernel techniques, the Fréchet–Montel principle, and approximation by dilations with a Kolmogorov–Riesz type compactness criterion. We also present several applications of our main results. Consequently, we recover and sharpen the results of Choe et al. [5] and Sharma and Sharma [22].
In [2] we considered a quadratically constrained quadratic programming problem in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Unfortunately, the proof of Theorem 3.3 in the above mentioned paper is incomplete, because [1, Thm 3.5] was mistakenly cited there. The statement of the theorem yet holds, but its proof needs to be completed. In the following we present an addendum to the proof of [2, Thm. 3.3].
In this paper, we first define the class 𝒮_g,γ^k+1(Ω ) of g-starlike mappings of real order γ ( γ∈ (0,1] ) on the bounded starlike circular domain Ω in ℂ^n and give the growth theorem for functions f∈𝒮_g,γ^k+1(Ω ) , where z=0 is a zero of order k+1 of f(z)-z and g is a Carathéodory function in 𝔻 . Next, as applications of the growth theorem, we will establish the distortion theorems of the Fréchet-derivative type for a subclass of 𝒮_g,γ^k+1(Ω ) with a weak restrictive condition. We also establish the distortion theorems of the Jacobi-determinant type for some subclasses of 𝒮_g,γ^k+1(Ω ) . Several important results can be obtained indirectly. Our works improve the classic distortion theorem of holomorphic functions from the case in ℂ to the case on the bounded starlike circular domain in ℂ^n .
We give an exact description of Calderón’s construction on a pair of cones of monotonically decreasing functions in weighted Lebesgue spaces. We also establish the interpolation and reiteration theorems for these cones.
