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

When the weight is of the form [math]\displaystyle{ \tfrac{1}{1+x^m} }[/math], the weighted space is called polynomial-weighted.

References

• Hazewinkel, Michiel, ed. (2001), "Weighted space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page