A generalized integral and conjugate functions

The author gives a descriptive definition of the $LG^*$-integral. The $LG^*$-integral extends the Lebesgue integral and coincides with it for nonnegative functions. For a function $f(x)$, $LG^*$-integrable on $[0,2\pi]$, the $LG^*$-Fourier series is defined and is almost everywhere $(C,1)$ summable to $f(x)$; the conjugate series is $(C,1)$ summable to $\widetilde f(x)$, which is also $LG^*$-integrable on $[0,2\pi]$, and is its $LG^*$-Fourier series.
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