scholarly journals Applicability of multiplicative renormalization method for a certain function

2008 ◽  
Vol 2 (3) ◽  
Author(s):  
Izumi Kubo ◽  
Hui-Hsiung Kuo ◽  
Suat Namli
Keyword(s):  
Multiplicative Renormalization ◽  
Renormalization Method

INTERPOLATION OF CHEBYSHEV POLYNOMIALS AND INTERACTING FOCK SPACES

2006 ◽  
Vol 09 (03) ◽  
pp. 361-371 ◽  
Author(s):  
IZUMI KUBO ◽  
HUI-HSIUNG KUO ◽  
SUAT NAMLI
Keyword(s):  
Orthogonal Polynomials ◽  
Anderson Model ◽  
Chebyshev Polynomials ◽  
Field Operator ◽  
Parameter Family ◽  
Probability Measures ◽  
Fock Spaces ◽  
Arcsine Distribution ◽  
Multiplicative Renormalization ◽  
Renormalization Method

We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

scholarly journals GENERATING FUNCTIONS OF CAUCHY–STIELTJES TYPE FOR ORTHOGONAL POLYNOMIALS

2009 ◽  
Vol 12 (01) ◽  
pp. 91-98 ◽  
Author(s):  
MAREK BOŻEJKO ◽  
NIZAR DEMNI
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Short Proof ◽  
Probabilistic Interpretation ◽  
Infinitely Divisible ◽  
The Family ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Stieltjes Type

We give a free probabilistic interpretation of the multiplicative renormalization method. As a byproduct, we give a short proof of the Asai–Kubo–Kuo problem on the characterization of the family of measures for which this method applies with h(x) = (1 - x)-1 which turns out to be the free Meixner family. We also give a representation of the Voiculescu transform of all free Meixner laws (even in the non-freely infinitely divisible case).

CLASSES OF MRM-APPLICABLE MEASURES FOR THE POWER FUNCTION OF GENERAL ORDER

2013 ◽  
Vol 16 (04) ◽  
pp. 1350028
Author(s):  
IZUMI KUBO ◽  
HUI-HSIUNG KUO ◽  
SUAT NAMLI
Keyword(s):  
Orthogonal Polynomials ◽  
Power Function ◽  
Generating Functions ◽  
The Other ◽  
Large Classes ◽  
Beta Distributions ◽  
General Order ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Special Classes

The authors have previously studied multiplicative renormalization method (MRM) for generating functions of orthogonal polynomials. In particular, they have determined all MRM-applicable measures for renormalizing functions h(x) = ex, h(x) = (1 - x)-κ, κ = 1/2, 1, 2. For the cases h(x) = ex and (1 - x)-1, there are very large classes of MRM-applicable measures. For the other two cases κ = 1/2, 2, MRM-applicable measures belong to special classes of a certain kind of beta distributions. In this paper, we determine all MRM-applicable measures for h(x) = (1 - x)-κ with κ ≠ 0, 1, 1/2.

Renormalization method for inhomogeneity correction of MR images

2001 ◽  
Author(s):  
Dongqing Chen ◽  
Lihong Li ◽  
Daeki Yoon ◽  
J. H. Lee ◽  
Zhengrong Liang
Keyword(s):  
Mr Images ◽  
Inhomogeneity Correction ◽  
Renormalization Method
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