Kontorovich-Lebedev-transform(2)

The integral transform

$$ F ( \tau ) = \ \int\limits _ { 0 } ^ \infty K _ {i \tau } ( x) f ( x) d x , $$

where $ K _ \nu $ is the Macdonald function.

If $ f $ is of bounded variation in a neighbourhood of a point $ x = x _ {0} > 0 $ and if

$$ f ( x) \mathop{\rm ln} x \in L \left ( 0 , \frac{1}{2} \right ) ,\ \ f ( x) \sqrt x \in L \left ( \frac{1}{2} , \infty \right ) , $$

then the following inversion formula holds:

$$ \frac{f ( x _ {0} + ) + f ( x _ {0} - ) }{2 } = $$

$$ = \ \frac{2}{\pi ^ {2} x _ {0} } \int\limits _ { 0 } ^ \infty K _ {i \tau } ( x _ {0} ) \tau \sinh \pi \tau F ( \tau ) d \tau . $$

Let $ f _ {i} $, $ i = 1 , 2 $, be real-valued functions with

$$ f _ {i} ( x) x ^ {-} 3/4 \in L ( 0 , \infty ) ,\ \ f _ {i} ( x) \in L _ {2} ( 0 , \infty ) ; $$

and let

$$ F _ {i} ( \tau ) = \ \int\limits _ { 0 } ^ \infty \frac{\sqrt {2 \tau \sinh \pi \tau } } \pi \frac{K _ {i \tau } }{\sqrt x } f _ {i} ( x) d x . $$

Then

$$ \int\limits _ { 0 } ^ \infty F _ {1} ( \tau ) F _ {2} ( \tau ) d \tau = \ \int\limits _ { 0 } ^ \infty f _ {1} ( x ) f _ {2} ( x) d x $$

(Parseval's identity).

The finite Kontorovich–Lebedev transform has the form

$$ F ( \tau ) = \ \frac{2 \pi \sinh \pi \tau }{\pi ^ {2} | I _ {i \alpha } ( \alpha ) | ^ {2} } \times $$

$$ \times \int\limits _ { 0 } ^ \alpha [ K _ {i \tau } ( \alpha ) I _ {i \tau } ( x) - I _ {i \tau } ( \alpha ) K _ {i \tau } ( x) ] f ( x) \frac{dx}{x} , $$

$ \tau > 0 $, where $ I _ \nu $ is the modified Bessel function (see [3]).

The study of such transforms was initiated by M.I. Kontorovich and N.N. Lebedev (see [1], [2]).

References

[1]	M.I. Kontorovich, N.N. Lebedev, "A method for the solution of problems in diffraction theory and related topics" Zh. Eksper. i. Toer. Fiz. , 8 : 10–11 (1938) pp. 1192–1206 (In Russian)
[2]	N.N. Lebedev, Dokl. Akad. Nauk SSSR , 52 : 5 (1945) pp. 395–398
[3]	Ya.S. Uflyand, E. Yushkova, Dokl. Akad. Nauk SSSR , 164 : 1 (1965) pp. 70–72
[4]	V.A. Ditkin, A.P. Prudnikov, "Integral transforms and operational calculus" , Pergamon (1965) (Translated from Russian)

Comments

A transform table for the Kontorovich–Lebedev transform can be found in [a1]. A treatment in some detail of the transform is in [a2].

References

[a1]	A. Erdelyi, W. Magnus, F. Oberhettinger, "Tables of integral transforms" , 1–2 , McGraw-Hill (1954) pp. Chapt. XII
[a2]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6