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)

Comments

See also Hilbert kernel; Hilbert transform.

References

[a1]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[a2]	B.L. Moiseiwitsch, "Integral equations" , Longman (1977)