Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 16, pp. 1-8 (2000)

Transition from the dyadic to the real nonperiodic Hardy space

S. Fridli

Department of Numerical Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/D, H-1117 Budapest, Hungary,

Abstract: The dyadic Hardy space plays a special role in Walsh analysis. Namely, it separates the $L^p[0,1)$ $(1<p\leq \infty )$ and the $L^1[0,1)$ spaces, and in many cases the results received for the $(1<p\leq \infty )$ case can be extended for the dyadic Hardy space but not for $L^1[0,1).$ The real nonperiodic Hardy space, which is wider than the dyadic one, is often employed in the trigonometric Fourier analysis. It is natural to ask whether the results proved for the dyadic Hardy space remain true for the real nonperiodic Hardy space. The idea behind this question is to make it possible to compare the corresponding results in the trigonometric and in the Walsh analysis. In this paper we provide a simple method for solving this problem for $\sigma$-sublinear functionals. Also, we study two well known sequences of functionals to demonstrate how our method works.

Keywords: Walsh-Fourier series, dyadic Hardy norm, real nonperiodic Hardy norm

Classification (MSC91): 42C10

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