Weighted space explained

In functional analysis, a weighted space is a space of functions under a weighted norm, which is a finite norm (or semi-norm) that involves multiplication by a particular function referred to as the weight.

Weights can be used to expand or reduce a space of considered functions. For example, in the space of functions from a set

U\subset R to R under the norm \| ⋅ \| _{U} defined by: \|f\| _{U=\sup} _{x\in} {|f(x)|} , functions that have infinity as a limit point are excluded. However, the weighted norm \|f\| = \sup _{x\in} {\left|f(x)\tfrac{ 1 }{ 1+ x ^{2}\right|}} is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm \|f\| = \sup _{x\in} {\left|f(x)( 1 + x ^{4)\right|}} is finite for many fewer functions.

When the weight is of the form

\tfrac{ 1 }{ 1+ x ^{m}} , the weighted space is called polynomial-weighted.^[1]

References

1. Walczak. Zbigniew. 2005. On the rate of convergence for some linear operators. Hiroshima Mathematical Journal. 35. 115–124. 10.32917/hmj/1150922488 .