Properties of Multi-Variable Positive Real Functions on Real--and Imaginary--Axis

Shun-ichi OKABE; Hiroshi OZAKI

Properties of Multi-Variable Positive Real Functions on Real--and Imaginary--Axis

Shun-ichi OKABE, Hiroshi OZAKI

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Usually, for the definitions on n-variable positive real functions, the properties on 2n-dimentional domain are used as the conditions of the definition, as examplified below. On the domain D {(p₁, p₂, , p_n)|Re p₁0, Re p₂0, , Re pn0)}, |Re w(p₁, p₂, , p_n) must be positive." In the present paper, the approach is simplified by the following two items.
(i) Fixing all variables except for one on the axis (real axis or imaginary axis), the problems on multi-variables are reduced to the one's on singlevariable.
(ii) The problems are discussed only on one-dimentional domain, that is, on the axes.
Simplifying the problems concerning with multi-variable positive real function by the two items stated above, the definitions of several classes of positive real function are presented. Clearly, these definitions may simplify extremely the verification of positive realness of any given function, and may give an effective means for the studies of multi-variable positive real function.

Publication: IEICE TRANSACTIONS on transactions Vol.E61-E No.9 pp.717-724

Publication Date: 1978/09/25

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Type of Manuscript: PAPER

Category: Circuit Theory

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Shun-ichi OKABE
Hiroshi OZAKI

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Shun-ichi OKABE, Hiroshi OZAKI, "Properties of Multi-Variable Positive Real Functions on Real--and Imaginary--Axis" in IEICE TRANSACTIONS on transactions, vol. E61-E, no. 9, pp. 717-724, September 1978, doi: .
Abstract: Usually, for the definitions on n-variable positive real functions, the properties on 2n-dimentional domain are used as the conditions of the definition, as examplified below. On the domain D {(p₁, p₂, , p_n)|Re p₁0, Re p₂0, , Re pn0)}, |Re w(p₁, p₂, , p_n) must be positive." In the present paper, the approach is simplified by the following two items.
(i) Fixing all variables except for one on the axis (real axis or imaginary axis), the problems on multi-variables are reduced to the one's on singlevariable.
(ii) The problems are discussed only on one-dimentional domain, that is, on the axes.
Simplifying the problems concerning with multi-variable positive real function by the two items stated above, the definitions of several classes of positive real function are presented. Clearly, these definitions may simplify extremely the verification of positive realness of any given function, and may give an effective means for the studies of multi-variable positive real function.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e61-e_9_717/_p

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@ARTICLE{e61-e_9_717,
author={Shun-ichi OKABE, Hiroshi OZAKI, },
journal={IEICE TRANSACTIONS on transactions},
title={Properties of Multi-Variable Positive Real Functions on Real--and Imaginary--Axis},
year={1978},
volume={E61-E},
number={9},
pages={717-724},
abstract={Usually, for the definitions on n-variable positive real functions, the properties on 2n-dimentional domain are used as the conditions of the definition, as examplified below. On the domain D {(p₁, p₂, , p_n)|Re p₁0, Re p₂0, , Re pn0)}, |Re w(p₁, p₂, , p_n) must be positive." In the present paper, the approach is simplified by the following two items.
(i) Fixing all variables except for one on the axis (real axis or imaginary axis), the problems on multi-variables are reduced to the one's on singlevariable.
(ii) The problems are discussed only on one-dimentional domain, that is, on the axes.
Simplifying the problems concerning with multi-variable positive real function by the two items stated above, the definitions of several classes of positive real function are presented. Clearly, these definitions may simplify extremely the verification of positive realness of any given function, and may give an effective means for the studies of multi-variable positive real function.},
keywords={},
doi={},
ISSN={},
month={September},}

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TY - JOUR
TI - Properties of Multi-Variable Positive Real Functions on Real--and Imaginary--Axis
T2 - IEICE TRANSACTIONS on transactions
SP - 717
EP - 724
AU - Shun-ichi OKABE
AU - Hiroshi OZAKI
PY - 1978
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E61-E
IS - 9
JA - IEICE TRANSACTIONS on transactions
Y1 - September 1978
AB - Usually, for the definitions on n-variable positive real functions, the properties on 2n-dimentional domain are used as the conditions of the definition, as examplified below. On the domain D {(p₁, p₂, , p_n)|Re p₁0, Re p₂0, , Re pn0)}, |Re w(p₁, p₂, , p_n) must be positive." In the present paper, the approach is simplified by the following two items.
(i) Fixing all variables except for one on the axis (real axis or imaginary axis), the problems on multi-variables are reduced to the one's on singlevariable.
(ii) The problems are discussed only on one-dimentional domain, that is, on the axes.
Simplifying the problems concerning with multi-variable positive real function by the two items stated above, the definitions of several classes of positive real function are presented. Clearly, these definitions may simplify extremely the verification of positive realness of any given function, and may give an effective means for the studies of multi-variable positive real function.
ER -