Conjugate trigonometric series
to the series
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The series
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These series are the real and imaginary parts, respectively, of the series
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where . The formula for the partial sums of the trigonometric series
conjugate to the Fourier series of
is
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where is the conjugate Dirichlet kernel. If
is a function of bounded variation on
, then a necessary and sufficient condition for the convergence of
at a point
is the existence of the conjugate function (see Conjugate function Section 3)
, and this is then the sum of the series
. If
is a summable function on
, then
can be summed almost-everywhere by the method
,
, and by the Abel–Poisson method, and the sum coincides almost-everywhere with the conjugate of
. If
is summable, then the conjugate series
is its Fourier series. The function
need not be summable; in the case of generalizations of the Lebesgue integral such as the
-integral and the Boks integral, the conjugate series
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) |
Conjugate trigonometric series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_trigonometric_series&oldid=13205