Sentence examples for by a finite sum from inspiring English sources

Numerical problems arise from singularities in the integrand and from representation of an infinite integral by a finite sum.

A Marcinkiewicz type inequality is useful when we need to estimate (L_{p}) norms of a trigonometric polynomials by a finite sum.

Such symbols exist for example in the case that (sigma (xi)) is represented by a finite sum, and (beta_{k}in{mathbb {Q}}).

Given N observed samples {x 1,x 2,…,x N }, the expectation in (8) can be approximated by a finite sum.

This shows that by replacing the expectation operator by a finite sum over the realizations x k(n), we get the approximate maximum likelihood estimator, only we average over the sub-vectors x k(n).

(8) Equation (8) shows that the polynomial reproducing kernel function (R_{y}^{m}(x)) not only can easily be constructed by a finite sum of basis functions, also this kernel function and the associated reproducing kernel Hilbert space (Pi_{w}^{m}[a,b]) can easily be updated by increasing m.

Recently Walczak[2] proposed and studied the Szász-Mirakyan operators by considering a finite sum.

By establishing a finite sum inequality based on quadratic terms, a new delay-dependent bounded real lemma (BRL) for delayed singular systems is derived.

Practical implementation requires the truncation of the exact boundary condition by approximating an infinite sum with a finite sum, and by terminating an infinite recursion relation.

Lemma 4.6 The inequality { 1 2 π ∫ 0 2 π | ∫ a b g ( x, t ) d t | r d x } 1 / r ≤ ∫ a b { 1 2 π ∫ 0 2 π | g ( t, x ) | r d x } 1 / r d t. is known as generalized Minkowski's inequality where the generalization is simply replacing a finite sum by a definite (Lebesgue) integral (see Chui [16]).

First, we recall the notion of a finite sum '⊕' defined by Butsan et al. [4].