ProductSpaceConstWeighting¶

class odl.space.pspace.ProductSpaceConstWeighting(constant, exponent=2.0)[source]¶

Constant weighting for ProductSpace.

Attributes

Methods

`dist`(self, x1, x2)	Calculate the constant-weighted distance between two elements.
`equiv`(self, other)	Test if other is an equivalent weighting.
`inner`(self, x1, x2)	Calculate the constant-weighted inner product of two elements.
`norm`(self, x)	Calculate the constant-weighted norm of an element.

__init__(self, constant, exponent=2.0)[source]¶

Initialize a new instance.

Parameters

constantpositive float: Weighting constant of the inner product
exponentpositive float, optional: Exponent of the norm. For values other than 2.0, no inner product is defined.

Notes

For exponent 2.0, a new weighted inner product with constant $c$ is defined as

$\langle x, y \rangle_c = c\, \langle x, y \rangle.$

For other exponents, only norm and `dist are defined. In the case of exponent inf, the weighted norm is

$\|x\|_{c,\infty} = c\, \|x\|_\infty,$

otherwise it is

$\|x\|_{c,p} = c^{1/p} \, \|x\|_p.$
Note that this definition does not fulfill the limit property in $p$ , i.e.,

$\|x\|_{c,p} \not\to \|x\|_{c,\infty} \quad \text{for } p \to \infty$

unless $c = 1$ . The reason for this choice is that the alternative with the limit property consists in ignoring the weight altogether.
The constant must be positive, otherwise it does not define an inner product or norm, respectively.