Conjugate Fourier series

In the 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. Template:Harvtxt studied the delicate questions of convergence of this series, and its relationship with the Hilbert transform.

In detail, consider a trigonometric series of the form

$f(\theta )=\frac{1}{2}{a}_0+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos n\theta +b_n\sin n\theta )$

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

$F(z)=\frac{1}{2}{a}_0+\sum _{n=1}^{\mathrm{\infty }}(a_n-i{b}_n)z^n$

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

$\tilde{f}(\theta )=\sum _{n=1}^{\mathrm{\infty }}(a_n\sin n\theta -b_n\cos n\theta ).$

See also

• Harmonic conjugate

References

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