View source for Conjugate trigonometric series

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''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|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|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| $  A $-
integral]] and the [[Boks integral|Boks integral]], the conjugate series  $  \widetilde \sigma   [ f] $
is always the Fourier series of the conjugate function.

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

====Comments====
Reference [[#References|[7]]] is a long useful survey. The references [[#References|[a1]]], [[#References|[a2]]] are standard.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1959–1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.H. Hardy,   W.W. Rogosinsky,   "Fourier series" , Cambridge Univ. Press  (1950)</TD></TR></table>