A remuneration list

In this paper we consider the evaluation of integrals of the following forms:

#math20#

#tex2html_wrap_indisplay1223##tex2html_wrap_indisplay1224##tex2html_wrap_indisplay1225#E_iB_{i, k, x}(t)#tex2html_wrap_indisplay1226##tex2html_wrap_indisplay1227##tex2html_wrap_indisplay1228#F_jB_{j, l, y}(t)#tex2html_wrap_indisplay1229#dt,

(1)

#math21#

#tex2html_wrap_indisplay1231#f (t)#tex2html_wrap_indisplay1232##tex2html_wrap_indisplay1233#E_iB_{i, k, x}(t)#tex2html_wrap_indisplay1234#dt,

(2)

where B_{i, k, x} is the ith B-spline of order k defined over the knots #math22#x_i, x_i+1,…, x_i+k. We will consider B-splines normalized so that their integral is one. The splines may be of different orders and defined on different knot sequences x and y. Often the limits of integration will be the entire real line, - ∞ to + ∞. Note that (#problem#76>) is a special case of (#problem2#77>) where f (t) is a spline.

There are five different methods for calculating (#problem#78>) that will be considered:
#remunerate79#
Of these five, only methods 1 and 5 are suitable for calculating (#problem2#82>). The first four methods will be touched on and the last will be discussed at length.