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

When the weight is of the form $\frac{1}{1+x^m}$, the weighted space is called polynomial-weighted.

References

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