Convergence and the Fourier character of trigonometric transforms with slowly varying convergence moduli (Q1898139)

For \(p> 1\), the convergence modulus of the trigonometric transform \(\sum_{|n|< \infty} c(n) e^{int}\) is defined in Math. Ann. 279, 103-115 (1987; Zbl 0625.42003) by the second author as \(k^p_n(c)= \sum_{|k|\leq n} |k|^{p- 1}|\Delta c(k)|^p\). The convergence modulus for \(c= \widehat f\), \(f\in L^1\), is a Tauberian device whose restriction in growth: (i) recovers \(L^1\)-convergence of the Fourier series to \(f\), (ii) recovers a.e. convergence of the Fourier series to \(f\) and (iii) determines the structure of the Fourier coefficients. Later [the second author, Proc. Third Annual Meeting Int. Workshop Anal. Appl., Inst. Math., Novi Sad 1-5 (1991)] an equivalent form of the convergence modulus is generalized to \(n^{{1\over p}}|\sigma_n(c)- S_n(c)|_p\), and results analogous to (i), (ii) and (iii) are proved. In this study, the authors develop a theory of convergence and integrability of the trigonometric transforms satisfying \[ {1\over n} \sum_{|k|\leq n} |k|^p|\Delta c(k)|^p= O(1),\quad n> 1, \quad p\in (1, 2],\tag{i} \] \[ n^{{1\over p}}|\sigma_n(c)- S_n(c)|_p= O(1),\quad n\to \infty,\quad p>1.\tag{ii} \] The results obtained follow a natural direction of inquiry and significantly extend previous results.