Weighted space

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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.

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

References

↑ Walczak, Zbigniew (2005). "On the rate of convergence for some linear operators". Hiroshima Mathematical Journal 35: 115–124.

Kudryavtsev, L D (2001). "Weighted Space". In Michiel Hazewinkel. Encyclopaedia of Mathematics. Springer.

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