Physical Inner Products
. The non-Hermiticity property of operators might cause complications in calculations.
Based on the independence of the random vector X11 and the random variables ([[xi].sub.j], p + 1 [less than or equal to] j [less than or equal to] n), the inner products
in (56) are rearranged as
To guarantee correctness, our scheme requires that inner products
be within a range of polynomial-size, which is consistent with other schemes in Table 1.
The following three combinations of S, [<<x,x>>.sub.S], and N, whose block inner products
were first deduced in , satisfy the above definitions.
Taking the inner products
<(1.1),2[phi]>, and <(1.2),2[psi]>, then summing up the resulting equalities and by the Neumann boundary conditions, we get
As a matter of fact, several inner products
are to be defined when considering the inverse optimization problem.
Furthermore, matching the inner products
of two Hermite polynomial tensors of equal rank, up to a given order N, will guarantee orthogonality of the Hermite polynomial tensors in the discrete space:
According to wavelet-Galerkin method, connection coefficients are the inner products
of Daubechies scaling functions and their derivatives, because we are taking Daubechies scaling functions as a Galerkin basis.
According to equation (1), all of the inner products
in Table 6 are shown as below:
This will be done by taking into account the weighting function in the inner products
and by examining these ones case by case.
Sanz-Serna: On polynomials orthogonal with respect to certain Sobolev inner products
The core of the book presents an axiomatic development of the most important elements of finite-dimensional linear algebra: vector spaces, linear operators, norms and inner products
, and determinants and eigenvalues.