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| {{Orphan|date=September 2011}}
| | I'm Nestor and I live with my husband and our 3 children in Laufen, in the south area. My hobbies are Sand castle building, Model Aircraft Hobbies and Magic.<br><br>Here is my web-site; [http://tinyurl.com/lg86kzh Celine Handbags UK] |
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| In the [[mathematics|mathematical]] field of [[Fourier analysis]], the '''conjugate Fourier series''' arises by realizing the Fourier series formally as the boundary values of the [[real part]] of a [[holomorphic function]] on the [[unit disc]]. The [[imaginary part]] of that function then defines the conjugate series. {{harvtxt|Zygmund|1968}} studied the delicate questions of convergence of this series, and its relationship with the [[Hilbert transform]].
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| In detail, consider a [[trigonometric series]] of the form
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| :<math>f(\theta) = \tfrac12 a_0 + \sum_{n=1}^\infty \left(a_n\cos n\theta + b_n\sin n\theta\right)</math>
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| in which the coefficients ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub> are [[real number]]s. This series is the real part of the [[power series]]
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| :<math>F(z) = \tfrac12 a_0 + \sum_{n=1}^\infty (a_n-ib_n)z^n</math> | |
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| along the [[unit circle]] with <math>z=e^{i\theta}</math>. The imaginary part of ''F''(''z'') is called the '''conjugate series''' of ''f'', and is denoted
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| :<math>\tilde{f}(\theta) = \sum_{n=1}^\infty \left(a_n\sin n\theta - b_n\cos n\theta\right).</math>
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| ==See also==
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| * [[Harmonic conjugate]]
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| ==References==
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| * {{Citation | last1=Grafakos | first1=Loukas | title=Classical Fourier analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-09431-1 | doi=10.1007/978-0-387-09432-8 | mr=2445437 | year=2008 | volume=249}}
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| *{{citation |title=Trigonometric series |first=Antoni |last=Zygmund |authorlink=Antoni Zygmund |publisher=Cambridge University Press |year=1968 |publication-date=1988 |isbn=978-0-521-35885-9 |edition=2nd}}
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| [[Category:Fourier analysis]]
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| [[Category:Fourier series]]
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| {{mathanalysis-stub}}
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Latest revision as of 01:38, 8 June 2014
I'm Nestor and I live with my husband and our 3 children in Laufen, in the south area. My hobbies are Sand castle building, Model Aircraft Hobbies and Magic.
Here is my web-site; Celine Handbags UK