Weight

weight function

A functional multiplier allowing one to obtain a finite norm of a given type for a function whose norm (or semi-norm), without this multiplier, would be infinite. The concept of a weight (function) plays an important part in problems concerning the approximation of functions (in particular, on unbounded intervals), in moment problems, in the theory of imbedding of function spaces (cf. Weighted space), in problems concerning the extension of functions, and in the theory of differential equations.

Comments

For instance, one consider on the space $ C[ a, b] $ of continuous functions on the interval $ [ a, b] $ the weighted square norm, [a2],

$$ \| f \| = \int\limits _ { a } ^ { b } w( x) f ^ { 2 } ( x) dx $$

where $ w: [ a, b] \rightarrow \mathbf R $ is a continuous function with positive values.

One says that polynomials $ P _ {j} ( x) $ on $ [ a, b] $ are orthonormal with respect to a weight $ w( x) $ if, [a2],

$$ \int\limits _ { a } ^ { b } w( x) P _ {j} ( x) P _ {i} ( x) dx = \delta _ {ij} . $$

Cf. also Weighted average for another illustration of the idea of assigning weights (different influence, importance) to the various constituents that make up a sum, integral or other mathematical construct.

Still other illustrations of the idea are weighted least squares, [a3], where, for example, an approximation or estimate is sought that minimizes a weighted sum

$$ \sum _ { i= } 1 ^ { n } w _ {i} \| \widehat{x} - x _ {i} \| ^ {2} , $$

and the weighted matching problem, where one considers a weighted graph (i.e. a graph with a weight specified for each edge) and the problem is to find a matching (i.e. a set of edges no two of which share a vertex) of maximal total weight, [a4].

In filtering and identification problems it is often a good idea to give more recent observations more weight than those in the far past. In this connection one speaks often of gains and gain sequences instead of weights, [a5], [a6].

In an integration formula

$$ \int\limits _ { a } ^ { b } w( x) f( x) dx = \sum _ { k= } 0 ^ { N } w _ {k} f ( x _ {k} ) + R $$

with prescribed weighting function $ w( x) $, the $ w _ {k} $ are called weighting coefficients, [a7].

A linear dynamical input-output system

$$ \dot{x} ( t) = A x ( t) + Bu ( t), $$

$$ y( t) = Cx( t) $$

gives rise to a relation

$$ y( t) = \int\limits _ { 0 } ^ { t } W( t- \tau ) u( \tau ) d \tau $$

between input and output functions (sometimes called the black-box representation), in which $ W( t) $ is called the transfer function or weighting function, [a8].

In optimization over a finite or infinite horizon, e.g. in the theory of optimal economic growth, one aims to maximize a future return function

$$ \int\limits _ { t _ {0} } ^ { {t _ 1 } } e ^ {- \lambda ( t- t _ {0} ) } U( c( t)) dt , $$

where $ U $ is a utility function and $ c( t) $ the stream of goods available for consumption. The weighting function $ \mathop{\rm exp} ( - \lambda t) $ used here is uniquely determined by the discount factor $ \lambda $, [a9].

References

[a1]	J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
[a2]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 49; 134
[a3]	P.S. Maybeck, "Stochastic models, estimation, and control" , 1 , Acad. Press (1979) pp. 120; 232
[a4]	C.H. Papadimitriou, K. Steiglitz, "Combinatorial optimization. Algorithms and complexity" , Prentice-Hall (1982) pp. 247ff
[a5]	L. Ljung, "Recursive identification" M. Hazewinkel (ed.) J.C. Willems (ed.) , Stochastic systems: the mathematics of filtering and identification , Reidel (1981) pp. 247–283; esp. 272
[a6]	P.S. Maybeck, "Stochastic models, estimation, and control" , III , Acad. Press (1982) pp. 242–244
[a7]	F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 209
[a8]	G.S.G. Beveridge, R.S. Schechter, "Optimization: theory and practice" , McGraw-Hill (1970) pp. 53
[a9]	M.D. Intrilligator, "Mathematical optimization and economic theory" , Prentice-Hall (1971) pp. 407