On the Parseval equality and the Dini-Lipschitz condition with respect to the Vilenkin system
Copyright policy )On the Parseval equality and the Dini-Lipschitz condition with respect to the Vilenkin system
The Parseval equality (*) \(\sum^{\infty}_{k=0}a_ k\bar l_ k=\) is true for the Fourier coefficients \(a_ k\), resp. \(l_ k\) of any elements f and g of a Hilbert space H with respect to any complete orthonormal system \(\Phi\) in H. If H stands for \(L^ 2(U)\) with the usual scalar product, where U is the so called Vilenkin group [N. Ya. Vilenkin, Izv. Akad. Nauk SSSR, Ser. Mat., 11, 363-400 (1947; Zbl 0036.356)] and \(\Phi\) is the character system of U, the author gives sufficient conditions for the functions \(f\in L^ 1(U)\) and \(g\in C(U)\) so that (*) will be true.
- Eötvös Loránd University Hungary
Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), Dini-Lipschitz condition, Inner product spaces and their generalizations, Hilbert spaces, Fourier coefficients, character system, Vilenkin group, Parseval equality
Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), Dini-Lipschitz condition, Inner product spaces and their generalizations, Hilbert spaces, Fourier coefficients, character system, Vilenkin group, Parseval equality
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