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A simple and elementary proof of the Kchintchine inequality with the best constant. (English) Zbl 0623.42015
In order to obtain the best possible constant in Kchintchine inequality S. J. Szarek [Stud. Math. 58, 197-208 (1976; Zbl 0424.42014)] proved the following inequality for \(p=1:\) \[ \sqrt{2}\int^{1}_{0}| \sum^{k}_{i=1}\gamma_ i(t)a_ i| dt\geq (\sum^{k}_{i=1}a^ 2_ i)^{1/2}, \] for \(k\in N\) and \(q_ 1,q_ 2,...,q_ k\in R\), where \(r_ i(t)\) is the i-th Rademacher’s function. He proved that \(\sqrt{2}\) is the best possible constant in this inequality. In the present paper the author provides a simpler and more elementary proof of this inequality.
Reviewer: S.M.Mazhar

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)