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. {{harvtxt|Zygmund|1968}} 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

:$f(\backslash theta)\; =\; \backslash tfrac12\; a\_0\; +\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash left(a\_n\backslash cos\; n\backslash theta\; +\; b\_n\backslash sin\; n\backslash theta\backslash right)$

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

:$F(z)\; =\; \backslash tfrac12\; a\_0\; +\; \backslash sum\_\{n=1\}^\backslash infty\; (a\_n-ib\_n)z^n$

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

:$\backslash tilde\{f\}(\backslash theta)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash left(a\_n\backslash sin\; n\backslash theta\; -\; b\_n\backslash cos\; n\backslash theta\backslash right).$