1.1 Classification of Inverse Problems 3
and scaling
G(αm) = αG(m). (1.6)
In the case of a discrete linear inverse problem, (1.4) can always be written in the form of
a linear system of algebraic equations. See Exercise 1.1.
G(m) = Gm = d. (1.7)
In a continuous linear inverse problem, G can often be expressed as a linear integral operator,
where (1.1) has the form
b
a
g(s, x) m(x) dx = d(s) (1.8)
and the function g(s, x) is called the kernel. The linearity of (1.8) is easily seen because
b
a
g(s, x)(m
1
(x) + m
2
(x)) dx =
b
a
g(s, x) m
1
(x) dx +
b
a
g(s, x)m
2
(x) dx (1.9)
and
b
a
g(s, x) αm(x) dx = α
b
a
g(s, x) m(x) dx. (1.10)
Equations in the form of (1.8), where m(x) is the unknown, are called Fredholm integral
equations of the first kind (IFKs). IFKs arise in a surprisingly large number of inverse
problems. A key property of these equations is that they have mathematical properties that
make it difficult to obtain useful solutions by straightforward methods.
In many cases the kernel in (1.8) can be written to depend explicitly on s − x, producing a
convolution equation
∞
−∞
g(s − x) m(x) dx = d(s). (1.11)
Here we have written the interval of integration as extending from minus infinity to plus
infinity, but other intervals can easily be accommodated by having g(s −x) = 0 outside of the
interval of interest. When a forward problem has the form of (1.11), determining d(s) from
m(x) is called convolution, and the inverse problem of determining m(x) from d(s) is called
deconvolution.
Another IFK arises in the problem of inverting a Fourier transform
(f ) =
∞
−∞
e
−ı2πf x
φ(x) dx (1.12)
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