Weighted space

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

References

• Kudryavtsev, L.D. (2001) [1994], "Weighted space", in Michiel Hazewinkel (ed.), Encyclopedia of Mathematics, EMS Press