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单词 Weighted space
释义

Weighted space

英语百科

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)x^4\right|} is finite for many fewer functions.

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