On the Independence Property

A function #tex2html_wrap_inline3043# satisfies IP <#927#> if only if, for every positive integer l, every set of non-zero real numbers #tex2html_wrap_inline3047#, and every set of real numbers #tex2html_wrap_inline3049# which satisfy

#equation1807#

the functions:

#equation1813#

are linearly independent<#927#>. This linear independence means that unless

#equation1816#

it must be that, for some u :

#equation1819#

Elements of equation (#tex2html_wrap_inline3053#) are of the linear space (over #tex2html_wrap_inline3055#), consisting of all functions defined on #tex2html_wrap_inline3057# :

#equation1822#

Otherwise, they are linearly dependent and one of them can be formed as a linear combination of the others. #tex2html_wrap_inline3059# In addition, #tex2html_wrap_inline3061# satisfies the <#629#> Weak Independence Property<#629#> (WIP), if the above IP is only required to hold for all pairs with #tex2html_wrap_inline3063#. There are many nonlinear maps that do not satisfy IP. An example of functions that belongs to this category are <#630#> polynomials<#630#>, <#631#> periodic functions<#631#>, and the <#632#> exponential function<#632#> [#4##1#].