Conjugate Fourier series

In the

In detail, consider a trigonometric series of the form

${\displaystyle f(\theta )={\tfrac {1}{2}}a_{0}+\sum _{n=1}^{\infty }\left(a_{n}\cos n\theta +b_{n}\sin n\theta \right)}$

in which the coefficients a_n and b_n are real numbers. This series is the real part of the power series

${\displaystyle F(z)={\tfrac {1}{2}}a_{0}+\sum _{n=1}^{\infty }(a_{n}-ib_{n})z^{n}}$

along the unit circle with ${\displaystyle z=e^{i\theta }}$. The imaginary part of F(z) is called the conjugate series of f, and is denoted

${\displaystyle {\tilde {f}}(\theta )=\sum _{n=1}^{\infty }\left(a_{n}\sin n\theta -b_{n}\cos n\theta \right).}$

See also

• Harmonic conjugate

References

• Grafakos, Loukas (2008), Classical Fourier analysis, Graduate Texts in Mathematics, vol. 249 (2nd ed.), Berlin, New York:
  MR 2445437
• ISBN 978-0-521-35885-9

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