$#C+1 = 24 : ~/encyclopedia/old_files/data/S083/S.0803480 Schur theorems
...ficient problem]] for bounded analytic functions. They were obtained by I. Schur [[#References|[1]]]. Let $ B $
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...define the Schur function $S_\lambda(X)$ combinatorially as the generating function of all semi-standard Young tableaux of shape $\lambda$ filled with indices
...ties in common with ordinary Schur functions. See [[#References|[a3]]] for Schur functions.
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...f ideals in the algebras $A(t)$ and the establishment of a criterion for a function $x=\{x(t)\}$ to belong to the algebra $A$. A more frequently considered cas
...algebras with an involution (the continuous analogue of the [[Schur lemma|Schur lemma]]).
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is a function in $ H ^ \infty $
The Schur method for solving the Carathéodory interpolation problem [[#References|[a
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...extension problem is to find (if possible) an [[Analytic function|analytic function]] $ f $
is an $ H ^ \infty $-function with $ \| f \| _ \infty \leq 1 $
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...le for linear characters (cf. [[Schur functions in algebraic combinatorics|Schur functions in algebraic combinatorics]]). In fact, both are special cases of
There are by now (as of 2000) several other definitions; the original by Schur [[#References|[a6]]] was in terms of Pfaffians (cf. [[Pfaffian|Pfaffian]]),
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...groups, by its Schur multiplier $M ( G )$ (cf. also [[Schur multiplicator|Schur multiplicator]]). A standard reference is [[#References|[a5]]].
...mmetric group|Symmetric group]]; [[Alternating group|Alternating group]]), Schur [[#References|[a11]]] further showed that
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...{ n + 1} = \Phi _ { n + 1 } ( 0 )$ is called a reflection coefficient or Schur or Szegö parameter.
where the Szegö function is defined as
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...the complex plane (cf. also [[Analytic function|Analytic function]]). The Schur class arises in diverse areas of classical analysis and operator theory, an
...z )$ in the Schur class. The numbers are defined in terms of a sequence of Schur functions which is constructed recursively by setting $S _ { 0 } ( z ) = S
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...urier–Stieltjes transform of a finite measure $\mu$ on $\hat { C }$ is the function $\hat{\mu}$ on $G$ defined by
...generalized Bochner theorem states that a [[Measurable function|measurable function]] on $G$ is equal, almost everywhere, to the Fourier–Stieltjes transform
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...m062160/m06216040.png" />, [[#References|[a2]]], [[#References|[a3]]]. The
Schur result, [[#References|[a1]]], can be reformulated to say that <img align="a
...ww.encyclopediaofmath.org/legacyimages/m/m062/m062160/m06216065.png" /> is
Schur convex if and only if it is symmetric, i.e. <img align="absmiddle" border="
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The generating function is:
...University of Chicago in 1896; see also [[#References|[a4]]]. In 1923, I. Schur [[#References|[a16]]] suggested that these polynomials be named in honour o
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The Schur functions $s_{ \lambda }$ are a special basis for the algebra of symmetric
...< j } ( x _ { i } - x _ { j } )$. In his thesis [[#References|[a11]]], I. Schur defined the functions which bear his name as
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...cter of a group|Character of a group]]), the associated generalized matrix function $d _ { \chi } ^ { G } : \mathbf{C} ^ { n \times n } \rightarrow \mathbf{C}$
...the Schur functions (cf. also [[Schur functions in algebraic combinatorics|Schur functions in algebraic combinatorics]]). The immanant $d _ { \chi _ { \lamb
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...associated a characteristic operator-valued function. This characteristic function reflects properties of the colligation; for instance, multiplication of col
is called an operator colligation, and the corresponding operator-valued function $W _ { \Theta } ( z )$ acting in $\mathfrak{C}$ and defined by
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where the so-called [[Schur complement]] matrix $\Delta$ is seen to be
It is often convenient to use generating-function language, and to define
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...te-dimensional representation is completely irreducible (see [[Schur lemma|Schur lemma]]), but an operator-irreducible finite-dimensional representation can
be a continuous function on $ G $
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...ns exist for the symmetric groups, and also for the alternating groups. I. Schur constructed equations for the alternating groups; it was shown, in particul
(partial sums of the expansion of the exponential function) have as Galois group the alternating group if $n\equiv0\pmod 4$, and the s
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...is finite, that $\operatorname{Ker} \pi$ is just the [[Schur multiplicator|Schur multiplicator]] of $G$, which was the motivation for Steinberg's study.
c) $\{ a , b \}$ is multiplicative as a function of $a$ or of $b$;
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...^ { 2 }$ is defined to be the operator of multiplication by the coordinate function $z$: $S: f ( z ) \rightarrow z f ( z )$. A closed linear subspace $\mathcal
...ly, of contractive analytic matrix-functions on the unit disc) in terms of Schur parameters (see [[#References|[a9]]] and [[#References|[a16]]]) has proved
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