Generalized Trapezoid Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces
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چکیده
Some generalized trapezoid type inequalities for the RiemannStieltjes integral of continuous complex valued integrands de ned on the complex unit circle C (0; 1) and various subclasses of integrators of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided. 1. Introduction In [13], in order to approximate the Riemann-Stieltjes integral R b a f (t) du (t) by the generalised trapezoid formula (1.1) [u (b) u (x)] f (b) + [u (x) u (a)] f (a) ; x 2 [a; b] the authors considered the error functional (1.2) T (f; u; a; b;x) := Z b a f (t) du (t) [u (b) u (x)] f (b) [u (x) u (a)] f (a) and proved that (1.3) jT (f; u; a; b;x)j H 1 2 (b a) + x a+ b 2 r b _ a (f) ; x 2 [a; b] ; provided that f : [a; b]! R is of bounded variation on [a; b] and u is of r H-Hölder type, that is, u : [a; b]! R satis es the condition ju (t) u (s)j H jt sj for any t; s 2 [a; b] ; where r 2 (0; 1] and H > 0 are given. The dual case, namely, when f is of q K Hölder type and u is of bounded variation has been considered by the authors in [7] in which they obtained the bound: jT (f; u; a; b;x)j (1.4) K " (x a) x _
منابع مشابه
Ostrowski’s Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces
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تاریخ انتشار 2013