# Hilbert singular integral

The improper integral (in the sense of the Cauchy principal value)

$$\widetilde{f} ( x) = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } f ( t) \mathop{\rm cotan} \frac{x - t }{2 } dt,$$

where the periodic function $f$ is called the density of the Hilbert singular integral, while $\mathop{\rm cotan} \{ {( x - t)/2 } \}$ is called its kernel. If $f$ is summable, $\widetilde{f}$ exists almost-everywhere; if $f$ satisfies the Lipschitz condition of order $\alpha$, $0 < \alpha < 1$, $\widetilde{f}$ exists for any $x$ and satisfies this condition as well. If $f$ has summable $p$- th power, $p > 1$, $\widetilde{f}$ has the same property, and

$$\left \{ \int\limits _ { 0 } ^ { {2 } \pi } | \widetilde{f} ( x) | ^ {p} dx \right \} ^ {1/p} \leq M _ {p} \left \{ \int\limits _ { 0 } ^ { {2 } \pi } | f ( x) | ^ {p} dx \right \} ^ {1/p} ,$$

where $M _ {p}$ is a constant independent of $f$. In addition, the inversion formula of Hilbert's singular integral,

$$f ( x) = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \widetilde{f} ( t) \mathop{\rm cotan} \frac{t - x }{2 } dt + { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } f ( t) dt ,$$

is valid. The function $\widetilde{f}$ is said to be conjugate with $f$.

#### References

 [1] D. Hilbert, "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint (1953) [2] M. Riesz, "Sur les fonctions conjugées" Math. Z. , 27 (1927) pp. 218–244 [3] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) [4] N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian)