Let us now consider an arbitrary function represented by its Fourier
decomposition. This function, denoted *F*(*x*), is a linear combination
of sinusoids as in Equation 6.22.

Since we have a way to solve for the quadratic bound for a canonical
sinusoid
at a contact point *z*, we should be able to
generalize this bound to all the individual terms in
Equation 6.22 using some bound properties such as
Equation 6.15. Each sinusoid in the linear combination
in Equation 6.22 is of the form
with *a*>0^{6.3}. In addition, we need to modify the contact point *z*with which we will look up the values of *w*, *k* and *y* since these
computations are particular to
and not
.

Figure 6.9(a) depicts the function
(*a*=2, *b*=3) which we wish to bound with a contact
point at *x*^{*}=*z*=-6. We instead compute the parabolic bound for
at
*x*=(2*(-6)+3)=-9 and obtain *w*=0.23, *y*=-10.98 and
*k*=0.49. We can thus insert those quantities into the parabola
expression and obtain the bound for the desired function
which is plotted in
Figure 6.9 (c).

Using Equation 6.14 we can further manipulate each term
by scaling it appropriately. Thus, we can get an expression for the
lower bound on *F*(*x*) in terms of a multitude of parabolic elements
(*p*_{2i}(*x*) and
*p*_{2i+1}(*x*)).

Since it is necessary to now maximize the above sum of bounds, we can
merely do so by taking the derivative of the sum with respect to *x*and setting that to 0. Maximizing the sum of quadratic terms thus
merely reduces to the solution of a linear system. This is not quite
as trivial as *choosing* the *y* locus as the maximum for a single
parabola but it is still extremely efficient. This operation is
depicted in Equation 6.25. The *x* value computed is
the locus of the maximum of the parabola. Thus, the equation
implements one maximization step and yields the operating or contact
point for the next iteration (for the bounding step). Repeating these
two steps will converge to a local maximum of the function as in
Figure 6.10.

Since we are able to optimize any combination of sinusoids with the
above, and any arbitrary (*l*2-bounded) analytic function can be
approximated by sinusoids (even in multiple dimensions), it is
possible in principle to use the above derivation to optimize a wide
variety of functions. These functions could also be non-analytic
functions which are only sampled and approximated by sinusoids using
least-squares. In addition, the above derivations for the canonical
sinusoid *sin*(*x*) function could have been done for another function
such as radial basis functions, *exp*(*x*), and so on and then
generalized to linear combinations of this canonical functions. One
small caveat exists when using linear combinations of arbitrary
bounded functions: the negative of the function is bounded differently
from the function itself. Thus, for arbitrary linear combinations, one
needs to have a bound for canonical *f*(*x*) *and* -*f*(*x*) (since a
bound flips from lower to upper bound due to the negative sign).