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On the application of maximum entropy to the moments problem. (English) Zbl 0763.41004

Reconstruction of a function \(f(x)\) from a finite set of its moments \(\int_ 0^ 1 x^ n f(x)dx\) when the condition of maximum entropy is imposed as a constraint was dealt with starting from imprecise data with errors specified by a \(\chi^ 2\)-function. The entropy for \(f(x)\) is defined as \[ S[p]=-\int_ 0^ 1 p(x)\ln p(x)dx,\quad\text{where}\quad p(x)=f(x)\left(\int_ 0^ 1 f(x)dx\right)^{-1}. \] The first \(N\) moments are \(\mu_ n=\int_ 0^ 1 x^ n p(x)dx\), \(n=1\) to \(N\), they are associated with errors \(\varepsilon_ n\). The solution \(p(x)\) is of the polynomial form \(p(x)=\exp(a_ 0+a_ 1 x+\dots+a_ n x^ n)\), the coefficients \(a_ 0,a_ 1,\dots,a_ n\) are found by a set of algebraic equations. An extension of the method is discussed by the first author, F. Genet and G. Mennesier [Phys. Rev. D 36, 3494-3501 (1987)].
Reviewer: V.Burjan (Praha)

MSC:

41A10 Approximation by polynomials
81V99 Applications of quantum theory to specific physical systems
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References:

[1] DOI: 10.1016/0021-9991(79)90102-5 · Zbl 0406.65034 · doi:10.1016/0021-9991(79)90102-5
[2] DOI: 10.1063/1.526446 · doi:10.1063/1.526446
[3] DOI: 10.1063/1.528812 · Zbl 0782.65010 · doi:10.1063/1.528812
[4] DOI: 10.1103/PhysRevD.36.3494 · doi:10.1103/PhysRevD.36.3494
[5] DOI: 10.1007/BF02902727 · doi:10.1007/BF02902727
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