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Convergence and the Fourier character of trigonometric transforms with slowly varying convergence moduli. (English) Zbl 0827.42003
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.

##### MSC:
 42A20 Convergence and absolute convergence of Fourier and trigonometric series
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##### References:
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