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Grüss-type inequalities. (English) Zbl 1007.26016

This paper discusses the connection between the Grüss inequality and the best approximation of functions by constants. Let \((X,\Sigma,\mu)\) be a probability space and let \(f,g\in L^{2}(\mu).\) Then \[ \left|\int_{X}fg d\mu-\int _{X}f d\mu\cdot\int_{X}g d\mu\right|\leq\inf\|f-\alpha\|_{L^{2}}\cdot\inf\|g-\beta\|_{L^{2}} \] and this fact is used to recover the discrete form of the Grüss inequality, as formulated by Biernacki, Pidek and Ryll-Nardzewski. Also, its connection with the original result of Grüss is presented in great detail. In the same spirit, Theorem 10 gives another Grüss-type inequality for the functions \(f\in L^{\infty}(\mu)\), which are orthogonal on all polynomials of degree \(\leq n:\) \[ \int_{X}|f|d\mu \leq\inf\{ \|f-P\|_{L^{\infty}} ;\text{ }P\text{ polynomial of degree }\leq n\}. \]

MSC:

26D15 Inequalities for sums, series and integrals
41A50 Best approximation, Chebyshev systems
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