# N-variable rational approximants and method of moments

## Abstract

The method of moments is applied to pairs linear permutable self-adjoint operators A and B in a Hilbert space H. An approximate expression for the diagonal matrix elements of the operator (1 - wA - zB)$sup -1$, where w, z are complex numbers, is taken as a guide to the definition of rational approximants from general formal power series in two variables. With an operator convergence theorem in a certain Hilbert space as a basis, the convergence of the approximants to analytic functions of two complex variables with the integral representation G(w,z) =integral$Integral$ d sigma($alpha$,$beta$)/(1 - w$alpha$ - z$beta$) is proved, under suitable restrictions on the positive measure sigma($alpha$,$beta$). The same approximation scheme can also be applied to the diagonal matrix elements of the operator [(1 - wA) (1 - zB)]$sup -1$, leading to a different rational approximant which is proved to converge to functions with the integral representation G(w,z) = $Integral$$Integral$ d sigma($alpha$,$beta$)/ (1 - w$alpha$) (1 - z$beta$). In both cases the convergence is uniform on appropriate compact subsets of C$sup 2$. The extension to the n-dimensional case is straightforward for both approximants. The connections with a standard variational principle are also briefly discussed. (auth)

- Authors:

- Publication Date:

- Research Org.:
- Stanford Univ., CA

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 4132888

- NSA Number:
- NSA-33-013819

- Resource Type:
- Journal Article

- Journal Name:
- J. Math. Phys. (N.Y.), v. 16, no. 4, pp. 840-845

- Additional Journal Information:
- Other Information: Orig. Receipt Date: 30-JUN-76

- Country of Publication:
- United States

- Language:
- English

- Subject:
- N76100* -Physics (Theoretical)-General; N76200 -Physics (Theoretical)-Quantum Field Theories; 644100*; *FIELD THEORIES- MATHEMATICS; *PADE APPROXIMATION; *QUANTUM MECHANICS- MATHEMATICS; HILBERT SPACE; MATHEMATICAL OPERATORS; MATRIX ELEMENTS; MOMENTS METHOD; POWER SERIES; SERIES EXPANSION

### Citation Formats

```
Alabiso, C., and Butera, P.
```*N-variable rational approximants and method of moments*. United States: N. p., 1975.
Web. doi:10.1063/1.522617.

```
Alabiso, C., & Butera, P.
```*N-variable rational approximants and method of moments*. United States. doi:10.1063/1.522617.

```
Alabiso, C., and Butera, P. Tue .
"N-variable rational approximants and method of moments". United States. doi:10.1063/1.522617.
```

```
@article{osti_4132888,
```

title = {N-variable rational approximants and method of moments},

author = {Alabiso, C. and Butera, P.},

abstractNote = {The method of moments is applied to pairs linear permutable self-adjoint operators A and B in a Hilbert space H. An approximate expression for the diagonal matrix elements of the operator (1 - wA - zB)$sup -1$, where w, z are complex numbers, is taken as a guide to the definition of rational approximants from general formal power series in two variables. With an operator convergence theorem in a certain Hilbert space as a basis, the convergence of the approximants to analytic functions of two complex variables with the integral representation G(w,z) =integral$Integral$ d sigma($alpha$,$beta$)/(1 - w$alpha$ - z$beta$) is proved, under suitable restrictions on the positive measure sigma($alpha$,$beta$). The same approximation scheme can also be applied to the diagonal matrix elements of the operator [(1 - wA) (1 - zB)]$sup -1$, leading to a different rational approximant which is proved to converge to functions with the integral representation G(w,z) = $Integral$$Integral$ d sigma($alpha$,$beta$)/ (1 - w$alpha$) (1 - z$beta$). In both cases the convergence is uniform on appropriate compact subsets of C$sup 2$. The extension to the n-dimensional case is straightforward for both approximants. The connections with a standard variational principle are also briefly discussed. (auth)},

doi = {10.1063/1.522617},

journal = {J. Math. Phys. (N.Y.), v. 16, no. 4, pp. 840-845},

number = ,

volume = ,

place = {United States},

year = {1975},

month = {4}

}