# Conjugate trigonometric series

to the series

$$\sigma = \ \frac{a _ {0} }{2} + \sum _ {n = 1 } ^ \infty a _ {n} \cos nx + b _ {n} \sin nx$$

The series

$$\widetilde \sigma = \ \sum _ {n = 1 } ^ \infty - b _ {n} \cos nx + a _ {n} \sin nx.$$

These series are the real and imaginary parts, respectively, of the series

$$\frac{a _ {0} }{2} + \sum _ {n = 1 } ^ \infty ( a _ {n} - ib _ {n} ) z ^ {n}$$

where $z = e ^ {ix}$. The formula for the partial sums of the trigonometric series $\widetilde \sigma [ f]$ conjugate to the Fourier series of $f$ is

$$\widetilde{S} _ {n} ( x) = \ { \frac{1} \pi } \int\limits _ {- \pi } ^ \pi f ( t) \widetilde{D} _ {n} ( t - x) dt,$$

where $\widetilde{D} _ {n} ( x)$ is the conjugate Dirichlet kernel. If $f$ is a function of bounded variation on $[- \pi , \pi ]$, then a necessary and sufficient condition for the convergence of $\widetilde \sigma [ f]$ at a point $x _ {0}$ is the existence of the conjugate function (see Conjugate function Section 3) $\widetilde{f} ( x _ {0} )$, and this is then the sum of the series $\widetilde \sigma [ f]$. If $f$ is a summable function on $[- \pi , \pi ]$, then $\widetilde \sigma [ f]$ can be summed almost-everywhere by the method $( C, \alpha )$, $\alpha > 0$, and by the Abel–Poisson method, and the sum coincides almost-everywhere with the conjugate of $f$. If $\widetilde{f}$ is summable, then the conjugate series $\widetilde \sigma [ f]$ is its Fourier series. The function $f$ need not be summable; in the case of generalizations of the Lebesgue integral such as the $A$- integral and the Boks integral, the conjugate series $\widetilde \sigma [ f]$ is always the Fourier series of the conjugate function.

#### References

 [1] A. Tauber, "Ueber den Zusammenhang des reellen und imaginären Teiles einer Potenzreihe" Monatsh. Math. Phys. , 2 (1891) pp. 79–118 [2] W.H. Young, Sitzungsber. Bayer. Akad. Wiss. München Math. Nat. Kl. , 41 (1911) pp. 361–371 [3] I.I. [I.I. Privalov] Priwalow, "Sur les fonctions conjuguées" Bull. Soc. Math. France , 44 (1916) pp. 100–103 [4] I.I. Privalov, "The Cauchy integral" , Saratov (1919) pp. 61–104 (In Russian) [5] N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1951) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) [6] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Oxford Univ. Press (1964) (Translated from Russian) [7] I.A. Vinogradova, "Generalized integrals and Fourier series" Itogi Nauk. Mat. Anal. 1970 (1971) pp. 65–107 (In Russian) [8] L.V. Zhizhiashvili, "Conjugate functions and trigonometric series" , Tbilisi (1969) (In Russian)