Pointwise

Summary

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value of some function An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise.

Pointwise operations

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Pointwise sum (upper plot, violet) and product (green) of the functions sin (lower plot, blue) and ln (red). The highlighted vertical slice shows the computation at the point x=2π.

Formal definition

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A binary operation o: Y × YY on a set Y can be lifted pointwise to an operation O: (XY) × (XY) → (XY) on the set XY of all functions from X to Y as follows: Given two functions f1: XY and f2: XY, define the function O(f1, f2): XY by

(O(f1, f2))(x) = o(f1(x), f2(x)) for all xX.

Commonly, o and O are denoted by the same symbol. A similar definition is used for unary operations o, and for operations of other arity.[citation needed]

Examples

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The pointwise addition   of two functions   and   with the same domain and codomain is defined by:

 

The pointwise product or pointwise multiplication is:

 

The pointwise product with a scalar is usually written with the scalar term first. Thus, when   is a scalar:

 

An example of an operation on functions which is not pointwise is convolution.

Properties

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Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. If   is some algebraic structure, the set of all functions   to the carrier set of   can be turned into an algebraic structure of the same type in an analogous way.

Componentwise operations

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Componentwise operations are usually defined on vectors, where vectors are elements of the set   for some natural number   and some field  . If we denote the  -th component of any vector   as  , then componentwise addition is  .

Componentwise operations can be defined on matrices. Matrix addition, where   is a componentwise operation while matrix multiplication is not.

A tuple can be regarded as a function, and a vector is a tuple. Therefore, any vector   corresponds to the function   such that  , and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.

Pointwise relations

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In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions AB can be ordered by fg if and only if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions AB with pointwise order.[1] Using the pointwise order on functions one can concisely define other important notions, for instance:[2]

An example of an infinitary pointwise relation is pointwise convergence of functions—a sequence of functions

 
with
 
converges pointwise to a function   if for each   in  
 

Notes

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  1. ^ Gierz et al., p. xxxiii
  2. ^ Gierz, et al., p. 26

References

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For order theory examples:

  • T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5.
  • G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.

This article incorporates material from Pointwise on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.