
Mathematical and Statistical Aspects of Inverse Problems 73
that the matrix J or
˜
J must be positive semi- de finite, so that the assump-
tion (A7) or (A7
′
) may not be overly restrictive. Alternatively, as has been
noted before, one can relax the assumptions (A7) or (A7
′
) by assuming the
existence of a dominating function b. Moreover, provided the weights w(t)
satisfy the requirement w(t) ≥ ¯w > 0 for all t ∈ [t
0
, t
f
], then one can use the
dominating function b (from the ordinary least squares problem) to obtain a
new dominating function
˜
b(E, t) = ¯w
−1
b(E, t) which is also ν-integrable. Even
in this case , though, one must still make the a dditio na