Conjugate function on the circle

Work on the torus $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ with the conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus: the characters are $e_k(x)=e^{2\pi ikx}$ for $k\in \mathbb{Z}$, the Fourier coefficients of a one-period integrable $f$ are $\widehat{f}(k)={\int }_0^{1}f(t)e^{-2\pi ikt}dt$, and a trigonometric polynomial is a finite complex linear combination of characters.

Let $f={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le M}{c}_k{e}_k$ be a trigonometric polynomial, so that $\widehat{f}(k)=c_k$ for $\mathrm{\mid }k\mathrm{\mid }\le M$ and $\widehat{f}(k)=0$ for $\mathrm{\mid }k\mathrm{\mid }>M$. The conjugate function of $f$ is the trigonometric polynomial

$Cf:={\sum }_{0<\mathrm{\mid }k\mathrm{\mid }\le M}(-i\mathrm{sgn}(k))\widehat{f}(k)e_k,\mathrm{sgn}(k):=\{\begin{array}{ll}1,& k>0,\\ 0,& k=0,\\ -1,& k<0.\end{array}$

Equivalently, $Cf$ is the unique trigonometric polynomial whose Fourier coefficients are

$\widehat{Cf}(k)=-i\mathrm{sgn}(k)\widehat{f}(k)(k\in \mathbb{Z})\mathrm{.}$

Consequently $\widehat{Cf}(0)=0$: every constant trigonometric polynomial lies in the kernel of $C$.

The assignment $C$ is complex-linear on the space of trigonometric polynomials. It also preserves real-valuedness: if $f$ is real, then $\widehat{f}(-k)=\overline{\widehat{f}(k)}$ for every $k$, and $-i\mathrm{sgn}(-k)\overline{\widehat{f}(k)}=\overline{-i\mathrm{sgn}(k)\widehat{f}(k)}$, so $\widehat{Cf}$ has the conjugate symmetry that characterizes a real trigonometric polynomial.

Two conventions are fixed by this definition. The factor $-i$ and the sign refer to the characters $e_k(x)=e^{2\pi ikx}$ and to the coefficient convention above; conjugating real functions, as in $f\mapsto -Cf$, is the opposite sign convention. And $C$ is defined on trigonometric polynomials alone: no bound on any $L^p(\mathbb{T})$ norm and no extension to arbitrary integrable functions is asserted here. The extension to $L^p(\mathbb{T})$ for $1<p<\mathrm{\infty }$ is proved on this page after the kernel formula below.