I investigate the convergence and summability of trigonometric and Walsh-Fourier series and trigonometric Fourier-transforms. Moreover, I consider the theory of classical Hardy spaces, dual spaces and interpolation spaces. The atomic decomposition of classical Hardy spaces is of basic importance in this theory. I generalize the theory to different functions spaces, too. I study similar questions for wavelet analysis and Gabor analysis as well. 

I consider martingale inequalities, martingale Hardy spaces, duality theorems as well as interpolation spaces. The atomic decomposition of the Hardy spaces is a key point in this theory. I study the theory of martingale Hardy spaces for different function spaces, too.