Approximation properties of beta operators
A localization theorem for beta approximation operators \(\beta_ n\) \((n=1,2,...)\) (introduced and discussed in the author's Thesis ''Study of some distributional and integral transforms'', Ph.D. Thesis, Kumaun University, Nainital, India, chap. IX): \(\beta_ n(f;x)=\int^{\infty}_{0}b_ n(x,u)f(u^{-1})du,\) where \(b_ n(x,u)=\frac{x^ n}{B(n,n)}\frac{u^{n-1}}{(1+xu)^{2n}},\) \(x>0\) is proved. With the help of this theorem the uniform convergence of \(\beta_ nf\) to f on every fixed interval \([x_ 1,x_ 2]\) \((0<x_ 1\leq x_ 2<\infty)\) is established. The main definitions and results are: Functional space M(0,\(\infty):\) M(0,\(\infty)\) is the linear space of the functions f(t) defined for \(t>0\), bounded and Lebesgue-measurable in every interval [r,R] \((0<r<R<\infty).\) Functional space H(0,\(\infty):\) H(0,\(\infty)\) is the linear space of the functions f(t)\(\in M(0,\infty)\) for which \(| f(t)| \leq Pt^{\alpha}\) \((P>0,\alpha >0,t>0).\) Theorem 2.1: Let f(x) and g(x) be two functions such that (i) f(x)\(\in H(0,\infty)\), (ii) g(x)\(\in M(0,\infty)\), (iii) f(x) and g(x) both are continuous at every point of the fixed interval \([x_ 1,x_ 2]\) \((0<x_ 1\leq x_ 2<\infty)\), (iv) \(f(x)=g(x)\) in \([x_ 1,x_ 2]\). Then there exists \(\beta_ n(f;x)\) for \(x\geq x_ 1\), \(n\geq [\alpha]+1\), and \(\lim_{n\to \infty}\beta_ n(f;x)=\lim_{n\to \infty}\beta_ n(g;x)\) holds uniformly in \([x_ 1,x_ 2].\) Theorem 2.2: Let f(x)\(\in H(0,\infty)\) be continuous at all points of the interval \([x_ 1,x_ 2]\) \((0<x_ 1\leq x_ 2<\infty)\). Then \(\beta_ n(f;x)\) exists for \(x\geq x_ 1\), \(n\geq [\alpha]+1\) and \(\lim_{n\to \infty}\beta_ n(f;x)=f(x)\) holds uniformly in \([x_ 1,x_ 2]\).
scientific article; zbMATH DE number 3929621