Summability of Multi-Dimensional Fourier Series and Hardy Spaces

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This monograph considers the theory of more-parameter dyadic and classical hardy spaces. It is a new application of martingale and distribution theories is dealt with. The theories of the multi-parameter dyadic martingale and the classical Hardy spaces are applied in Fourier analysis. Several summability methods of d-dimensional trigonometric-, walsh-, spline-, and ciesielski-series and fourier transforms as well as the d-dimensional dyadic derivative are investigated. The boundedness of the maximal operators of the summations on hardy spaces, weak (L1, L1) inequalities and a.e. convergence results for the d-dimensional fourier series are proved.