de Branges L., Rovnyak J. — Square summable power series :: Электронная библиотека попечительского совета мехмата МГУ

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de Branges L., Rovnyak J. — Square summable power series

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Название: Square summable power series

Авторы: de Branges L., Rovnyak J.

Язык:

Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1966

Количество страниц: 104

Добавлена в каталог: 02.04.2005

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Предметный указатель

$B^{\ast}(z)$       47
      91
$L^2(\mu)$       88—91
$\mathcal{C}(z)$       4—5 38 42
$\mathcal{H}(B)$       22—24 33
$\mathcal{H}(B)$ , characterization of spaces      38—39
$\mathcal{H}(B)$ , decomposition of spaces      29 32
$\mathcal{H}(B)$ , equality of spaces      33
$\mathcal{H}(B)$ , finite dimensional spaces      29 51 53—54
$\mathcal{H}(B)$ , inclusions      29—30 32—33
$\mathcal{H}(B)$ , relation to $\mathcal{H}(B^{\ast})$       47—49
$\mathcal{H}(B)$ , relation to $\mathcal{M}(B)$       23 29
$\mathcal{M}(B)$       11
Absolute value      1
Adjoint transformation      34 35 38 39 42
Adjoint transformation of difference-quotient transformation      36 37 39 46 47
Approximation by Blaschke products      51
Approximation by Borel measurable functions      86
Approximation by finite dimensional spaces      53
Approximation by polynomials      19
Aronszajn, N.      71
B-norm      22
Bessel’s inequality      7
Beurling, A.      12 34
Beurling’s factorization      97
Blaschke product      17 33 51 54
Borel measurable function      86—87
Borel measurable function, conjugate of      86
Borel measurable function, limits of      86
Borel measure      82—86
Borel measure and nondecreasing functions      82
Borel set      81
Bound of a transformation      47—48
Boundary value function      91—95 98
Bounded function      17 18—19 22 28 29 31—33 70
Bounded type      95—99
Bounded type and non-negative real part      96
Bounded type and square summable power series      96 98
Canonical model      34—35 42
Cantor diagonal procedure      53 80 81
Caratheodory condition      84
Cauchy sequence of complex numbers      1
Cauchy sequence of vectors      8
Cauchy’s Theorem      63—67
Change of measure theorem      79
Characterization of $\mathcal{C}(z)$       38
Characterization of difference-quotient transformation      34—35
Characterization of functions of bounded type      96
Characterization of range of multiplication by B(z)      29
Characterization of spaces $\mathcal{H}(z)$       39
Closed curve      62—63
Closed ideal      10—14 33
Closed ideal, examples      11 15 16 17 33
Closed ideal, generated by a power series      33 97 98
Closed set      8
Closed span      10
Commuting transformations      28 55—56
Completeness of $\mathcal{C}(z)$       8
Completeness of $\mathcal{H}(z)$       24
Completeness of complex numbers      2
Completeness of inner product spaces      8
Completeness of Lebesgue spaces      88
complex numbers      1
Conjugate of a Borel measurable function      86
Conjugate of a complex number      1
Construction of closed ideals      10—11
Construction of measures      83
Continuity of function values      14 28
Continuity of inner product      10
Continuity of length of a curve      59 63
Continuous linear functional      10
Continuous transformation      34
Contour integral      58
Convergent power series      14 18—19
Convergent sequence      1
Convex region      67
Convex set      8
Countable number of discontinuities      81
Curve      57
Decomposition of open sets      82
Decomposition of spaces $\mathcal{H}(B)$       29 32
Density of continuous functions in $L^2(\mu)$       91
Difference quotient      14 29
Difference quotient and minimal decompositions      36 44—46
Difference quotient in $\mathcal{H}(B)$       29 39 43
Difference quotient, identity      38—39 42—43 46
Difference-quotient transformation      34—39 45—47
Differentiability      14 57 59 67 68 70
Differentiable function      14 57
Differentiable function as derivative      67
Discontinuities of nondecreasing function      81
Disjoint sequence of sets      82
Double orthogonal complement      10
Equivalence classes of functions      88
Estimate of function values      97—99
Existence of factors      53
Existence of integral      58
Existence of subsequence      53 80
Existence of subspaces      53
Exponential function      3 52 55
Factorization      17 18 30—31 53 97—99
Factorization theorem      53
Factors of B(z)      17 18 29 30
Fatou’s theorem      91
Finite dimensional space      51 54
Formal convergence      19
Formal power series      2—3
Function values      14 28
Function values, bounded by one      17 18 70
Function values, estimate of      97—99
Fundamental theorem of algebra      18 70
Fundamental theorem of the calculus      59
Goursat, E.      63
Helly selection principle      80
Helly, E.      80
Herglotz, G.      79
Herglotz’s theorem      80
Hilbert space      8
Hilbert space of power series      38 39
Ideal      10
Ideal theorem      12
Ideal, closed      11—14 33
Ideal, examples of closed ideals      11 15 16 17 33

Identity for difference quotients      38—39 42—43 46
Inclusions of spaces $\mathcal{H}(B)$       29 30 32—33
Indeterminate      57
Infinite products      15
Infinite products, expressed in terms of norms      7 24
Infinite products, inner product      3 4
integral      58 79
Integral and sums      79
Integral over a curve      58—67
Integral, Lebesgue      87—91
Integral, Stieltjes      79 91
Integral, Stieltjes and Lebesgue      91
Interval of parametrization      57
Invariant subspace      33
Isometric inclusion      32
Isometric multiplication      11 32 33
Isometric multiplication and factors      33
Isometric substitution      49 54
Isometric transformation      34 47
Jensen’s Inequality      96—97
Kernel of a linear functional      10
Laguerre polynomial      3
Landau, E.      1
Lebesgue and Stieltjes integrals      91
Lebesgue integral      87—91
Lebesgue space      88
Lebesgue, H.      83
Left and right limits      59 64
Limits of Borel measurable functions      86
Linear functional      10
Linear transformation      2
Linearity      4
Logarithm      96
Maximum principle      17 29 68—70
Measurable function      86—87
Measure      82—86
Measure and nondecreasing function      82 83
Measure, change of measure theorem      79
Mesh of a partition      58
Minima of modulus      70
Minimal decomposition      26—29 31—32 36 44—46
Minimal decomposition and difference quotients      36 44 45
Multiplication by B(z)      11 18 28 29 33 54—55
Multiplication by z      13—14 38 43 45
Nevanlinna, R.      95 98—99
Nevanlinna’s estimate      98
Nevanlinna’s factorization      99
Non-negative real part      80
Non-negative real part and bounded type      96
Nondecreasing function      79 80 82 83 89
Norm      5
Norm of limit      19 91
One-dimensional range      34 37 39 42
Orthogonal complement      10
Orthogonal complement of ideal      23
Orthogonal sequence      7
Orthogonality      7
Outer measure      82—86
Parallelogram law      5 6—7 23
PARAMETER      57
Partial isometry      39 56
Partition      57
Perpendicular      7
Perpendicular and shortest distance      9
Poisson formula      76
Poisson kernel      92
Poisson representation      79
Polynomial      19
Polynomial and Blaschke products      33
Positive-definiteness      70
Positivity      4
Power series      see “Formal power series”
Power series, convergent power series      14 18—19
Power series, square summable power series      4
Powers tending to zero      34—35
Products of Borel measurable functions      86
Products of functions of bounded type      96
Products of power series      3
Projection      9
R(w)      36 46
Rational function and Blaschke product      33
Rectangular region      63
Refinement of a partition      58
Region      57
Representation of continuous linear functionals      10
Representation of differentiable function as power series      70
Riesz — Fischer theorem      88—91
Riesz, F., and Sz.-Nagy, B.      88
Rovnyak, J.      12
Schwarz inequality      5—6
Schwarz inequality, equality in      6 12
Schwarz’s lemma      70
Sequence of finite Blaschke products      51
Sequence of nondecreasing functions      80
Sequence of polynomials      19
Sequence of spaces      53
Series      see “Formal power series”
Series and functions      14 57
Sets of measure zero      88
Shortest distance      9
Soap bubble theorem      77
Space of dimension zero or one      53
span      9
Square integrable function      87—88
Square summable power series      4
Square summable power series and bounded type      96 98
Stieltjes and Lebesgue integrals      91
Stieltjes integral      79
Subadditivity of outer measure      82—83
Substitution theory      49—51 54—55
Sums of Borel measurable functions      86
Sums of functions of bounded type      96
Sums of squares bounded by one      22
Symmetry      4
Transformations      2 33—36 42
Transformations, commuting      28 55—56
Triangle inequality      6
Uniqueness of shortest distance      9
Unitary equivalence      34 42 47
Variable      57
Vector space      2
Vector space with inner product      4
Walsh, J. L.      15
zeros      17—18

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