Self-similar structures and fractal transforms in approximation theory

@article{Yukalov2002SelfsimilarSA,
  title={Self-similar structures and fractal transforms in approximation theory},
  author={Vyacheslav I. Yukalov and E. P. Yukalova},
  journal={Chaos Solitons \& Fractals},
  year={2002},
  volume={14},
  pages={839-861}
}
Interplay between Approximation Theory and Renormalization Group
  • V. Yukalov
  • Physics
    Physics of Particles and Nuclei
  • 2019
The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing
Self-similar extrapolation in quantum field theory
Calculations in field theory are usually accomplished by employing some variants of perturbation theory, for instance using loop expansions. These calculations result in asymptotic series in powers
From Asymptotic Series to Self-Similar Approximants
The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to
Calculation of critical exponents by self-similar factor approximants
Abstract.The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric ϕ4 theory and of the Ising glass. It is demonstrated that this method,
Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit
Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even
Extrapolation of perturbation-theory expansions by self-similar approximants
The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on
Extrapolation and interpolation of asymptotic series by self-similar approximants
The problem of extrapolation and interpolation of asymptotic series is considered. Several new variants of improving the accuracy of the self-similar approximants are suggested. The methods are
Self-similar variational perturbation theory for critical exponents.
  • H. Kleinert, V. Yukalov
  • Physics, Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2005
TLDR
From only three-loop perturbation expansions in 4-epsilon dimensions, analytic results for the exponents are obtained, which are close to those derived recently from ordinary field-theoretic variational perturbational theory to seventh order.
Describing phase transitions in field theory by self-similar approximants
Self-similar approximation theory is shown to be a powerful tool for describing phase transitions in quantum field theory. Self-similar approximants present the extrapolation of asymptotic series in
...
1
2
3
4
...

References

SHOWING 1-10 OF 179 REFERENCES
SELF-SIMILAR INTERPOLATION IN QUANTUM MECHANICS
An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar
Self-Similar Perturbation Theory
Abstract A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calculational algorithm can be derived. The
CRITICAL INDICES AS LIMITS OF CONTROL FUNCTIONS
A variant of self-similar approximation theory is suggested, permitting an easy and accurate summation of divergent series consisting of only a few terms. The method is based on a power-law algebraic
Degenerate trajectories and Hamiltonian envelopes in the method of self-similar approximations
We study two new techniques for the approximate calculation of the eigenvalues of the Schrodinger equation. These techniques are variants of the method of self-similar approximations suggested
Self-similar exponential approximants
An approach is suggested for defining effective sums of divergent series in the form of self-similar exponential approximants. The procedure of constructing these approximants from divergent series
...
1
2
3
4
5
...