Untersuchungen �ber ein Problem, das drei positiv definite quadratische Formen mit Streckenkomplexen in Beziehung setzt

Cauer, Wilhelm

doi:10.1007/bf01455810

Cited by 14 publications

(9 citation statements)

References 2 publications

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“…Analyticity is established from results of S. Bernstein [l] and R. P. Boas [2]. A Stieltjes integral representation of monotone operator functions is obtained using the Hamburger moment problem (4). This immediately gives analyticity in the entire complex plane off the real axis with the known mapping property.…”

Section: Introductionmentioning

confidence: 99%

Monotone and convex operator functions

Bendat¹,

Sherman²

1955

Trans. Amer. Math. Soc.

169

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Introduction.Monotone matrix functions of arbitrarily high order were introduced by Charles Loewner in the year 1934 [9](3) while studying realvalued functions which are analytic in their domain of definition, and continued in the complex domain, are regular in the entire open upper half-plane with non-negative imaginary part. The monotone matrix functions of order re defined later in the introduction deal with functions of matrices whose independent and dependent variables are real symmetric matrices of order re; monotone of arbitrarily high order means monotone for each finite integer re. The class corresponding to ra = 1 represents the functions which are monotone in the ordinary sense. Monotone operator functions are precisely the class of monotone matrix functions of arbitrarily high order. Loewner showed that analyticity plus the property of mapping the complex open upper halfplane into itself is characteristic for the class of monotone matrix functions of arbitrarily high order. Literature on this subject and its by-products, convex matrix functions and matrix functions of bounded variation, consists of but four papers, two by Loewner [9; 10 ] and one each from two of his doctoral students, F. Krauss [8] and O. Dobsch [5].The principal objective in this work was to discover whether convex operator functions (i.e. convex matrix functions of arbitrarily high order) satisfy properties analogous to those known for monotone operator functions. From the classical case it might be thought if a function is convex, its derivative should be monotone.

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Section: Introductionmentioning

confidence: 99%

Monotone and convex operator functions

Bendat¹,

Sherman²

1955

Trans. Amer. Math. Soc.

169

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“…Cauer (22) proceeds further and shows that (when considering (2) as an impedance function Z (>.» by the use of a continued fraction expansion (finite Stieltjes (39) continued fraction) we can obtain two more networks equivalent to those in Figs. 7 and 8 respectively in form of the two ladder structures shown in Fig.…”

Section: Charles M: Son Gewertzmentioning

confidence: 95%

“…13. 20 Cauer (22) also solves the case when the prescribed impedance function corresponds to a dissipative two-terminal network haz l -.0 Note, a partial fraction expansion of (2) and its reciprocal gives different networks, while a continued fraction expansion gives the same network.…”

Section: Charles M: Son Gewertzmentioning

confidence: 98%

“…From (22) it is clear that this ratio is simply the General Circuit Parameter D, and when considering the a:s (say) we see that the prescribed functions are: all, ann and ~.…”

mentioning

confidence: 97%

“…Again using (22) Then considering the whole right half of lhe A-plane and splitting up all a: S into their real and imaginary parts (denoted in a corresponding manner as the a:S themselves) according to (47), we get:…”

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confidence: 99%

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