Mutual kernels

reciprocal kernels

Two functions $ K ( x, s) $ and $ K _ {1} ( x, s) $ of real variables $ x, s $( or, in general, of points $ P $ and $ Q $ of a Euclidean space), defined on the square $ [ a, b] \times [ a, b] $ and satisfying the condition

$$ K _ {1} ( x, s) - K ( x, s) = $$

$$ = \ \int\limits _ { a } ^ { b } K ( x, t) K _ {1} ( t, s) dt = \int\limits _ { a } ^ { b } K _ {1} ( x, t) K ( t, s) dt. $$

If a kernel $ K _ {1} ( x, s) $ reciprocal with $ K ( x, s) $ exists, then $ K _ {1} ( x, s) $ is the resolvent kernel of the integral Fredholm equation

$$ \tag{* } \phi ( x) - \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x). $$

Comments

Indeed, when $ K( x, s) $ and $ K _ {1} ( x, s) $ are reciprocal kernels, the solution of equation (*) above is given by

$$ \phi ( x) = f( x) + \int\limits _ { a } ^ { b } K _ {1} ( x, t) f( t) dt. $$

Consider the Fredholm equation

$$ \tag{a1 } \phi ( x) = f( x) + \lambda \int\limits _ { a } ^ { b } K ( x, t) \phi ( t) dt $$

and the iterated kernels $ K ^ {(1)} ( x, t) = K ( x, t) $,

$$ K ^ {(n)} ( x, t) = \int\limits _ { a } ^ { b } K ^ {(n-1)} ( x, s) K( s, t) ds, \ n = 2, 3 ,\dots. $$

Form the Neumann series

$$ R( x, t; \lambda ) = K ^ {(1)} ( x, t) + \lambda K ^ {(2)} ( x, t) + \dots = $$

$$ = \ \sum _ {n= 1 } ^ \infty \lambda ^ {n-1} K ^ {(n)}( x, t). $$

If $ K( x, t) $ is continuous on $ [ a, b] \times [ a, b] $, this series is uniformly convergent for $ \lambda $ small. Then $ R( x, t; \lambda ) $ satisfies

$$ \lambda \int\limits _ { a } ^ { b } R( x, t; \lambda ) K( t, s) dt = R ( x, s; \lambda ) - K( x, s), $$

and

$$ \phi ( x) = f( x) + \lambda \int\limits _ { a } ^ { b } R ( x, t; \lambda ) f ( t) dt $$

solves (a1).

The terminology "mutual kernels" and "reciprocal kernels" is rarely used.

References

[a1]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[a2]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)
[a3]	B.L. Moiseiwitsch, "Integral equations" , Longman (1977)