On a theorem of Bellman on Fourier coefficients

In 1928 Hardy [1] proved that the class $L^p$ $(1\leqslant p<\infty)$ is invariant under the $(C,1)$-transformation of the Fourier coefficients. In 1944 Bellman [3] proved the dual result for the class $L^p$ $(1\leqslant p<\infty)$ with respect to the adjoint transformation of the Fourier coefficients the transpose of the matrix of the $(C,1)$-method.
In the present paper a new proof of Bellman's theorem that does not depend on Hardy's theorem is given, and a representation of the function with the transformed Fourier series in terms of the original function, similar to the Hardy representation, is obtained. In addition an inaccuracy in the statement of the second half of Bellman's theorem is corrected. Finally, integral analogues of these results are proved. These analogues were derived on a heuristic level in the paper of Bellman without justifying the computations and without stating the conditions imposed on the functions.