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''to the series
 
''to the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250701.png" /></td> </tr></table>
+
$$
 +
\sigma  = \
 +
 
 +
\frac{a _ {0} }{2}
 +
+
 +
\sum _ {n = 1 } ^  \infty 
 +
a _ {n}  \cos  nx +
 +
b _ {n}  \sin  nx
 +
$$
  
 
''
 
''
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The series
 
The series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250702.png" /></td> </tr></table>
+
$$
 +
\widetilde \sigma    = \
 +
\sum _ {n = 1 } ^  \infty 
 +
- b _ {n}  \cos  nx +
 +
a _ {n}  \sin  nx.
 +
$$
  
 
These series are the real and imaginary parts, respectively, of the series
 
These series are the real and imaginary parts, respectively, of the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250703.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{a _ {0} }{2}
 +
+
 +
\sum _ {n = 1 } ^  \infty 
 +
( a _ {n} - ib _ {n} )  z  ^ {n}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250704.png" />. The formula for the partial sums of the trigonometric series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250705.png" /> conjugate to the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250706.png" /> is
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250707.png" /></td> </tr></table>
+
$$
 +
\widetilde{S}  _ {n} ( x)  = \
 +
{
 +
\frac{1} \pi
 +
}
 +
\int\limits _ {- \pi } ^  \pi 
 +
f ( t) \widetilde{D}  _ {n} ( t - x)  dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250708.png" /> is the conjugate [[Dirichlet kernel|Dirichlet kernel]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c0250709.png" /> is a function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507010.png" />, then a necessary and sufficient condition for the convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507011.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507012.png" /> is the existence of the conjugate function (see [[Conjugate function|Conjugate function]] Section 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507013.png" />, and this is then the sum of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507015.png" /> is a summable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507017.png" /> can be summed almost-everywhere by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507019.png" />, and by the Abel–Poisson method, and the sum coincides almost-everywhere with the conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507021.png" /> is summable, then the conjugate series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507022.png" /> is its Fourier series. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507023.png" /> need not be summable; in the case of generalizations of the Lebesgue integral such as the [[A-integral|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507024.png" />-integral]] and the [[Boks integral|Boks integral]], the conjugate series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025070/c02507025.png" /> is always the Fourier series of the conjugate function.
+
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====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 17:46, 4 June 2020


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)

Comments

Reference [7] is a long useful survey. The references [a1], [a2] are standard.

References

[a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1959–1968)
[a2] G.H. Hardy, W.W. Rogosinsky, "Fourier series" , Cambridge Univ. Press (1950)
How to Cite This Entry:
Conjugate trigonometric series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_trigonometric_series&oldid=46473
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article