Alexander Ostrowski
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1 Ostrowski p. 1/3 Alexander Ostrowski Walter Gautschi Purdue University
2 Ostrowski p. 2/3 Collected Mathematical Papers Volume 1 Determinants Linear Algebra Algebraic Equations
3 Ostrowski p. 2/3 Collected Mathematical Papers Volume 1 Determinants Linear Algebra Algebraic Equations Volume 2 Multivariate Algebra Formal Algebra
4 Ostrowski p. 2/3 Collected Mathematical Papers Volume 1 Determinants Linear Algebra Algebraic Equations Volume 2 Multivariate Algebra Formal Algebra Volume 3 Number Theory Geometry Topology Convergence
5 Ostrowski p. 3/3 Collected Mathematical Papers Volume 4 Real Function Theory Differential Equations Differential Transformations (continued)
6 Ostrowski p. 3/3 Collected Mathematical Papers Volume 4 Real Function Theory Differential Equations Differential Transformations Volume 5 Complex Function Theory (continued)
7 Ostrowski p. 3/3 Collected Mathematical Papers Volume 4 Real Function Theory Differential Equations Differential Transformations Volume 5 Complex Function Theory Volume 6 Conformal Mapping Numerical Analysis Miscellaneous (continued)
8 Ostrowski p. 4/3 Ostrowski s mother Ostrowski, the cadett
9 Ostrowski p. 5/3 Vol.1. Determinants, Linear Algebra, Algebraic Equations Determinants and Linear Algebra (1937) Matrices with dominant diagonal A = [a ij ], d i := a ii j i a ij > 0, all i Hadamard (1899): det A 0 Ostrowski: det A i d i
10 Vol.1. Determinants, Linear Algebra, Algebraic Equations Determinants and Linear Algebra (1937) Matrices with dominant diagonal A = [a ij ], d i := a ii j i a ij > 0, all i Hadamard (1899): det A 0 Ostrowski: det A i d i (1937) M-matrices A = [a ij ], a ii > 0, a ij 0 (i j) a a 11 > 0, 11 a 12 a 21 a 22 > 0,..., det A > 0 Ostrowski p. 5/3
11 Ostrowski p. 6/3 Vol.1 (continued) Thm. If A is an M-matrix, then A 1 0 (1955) General theory of vector and matrix norms ( ) Rayleigh quotient iteration for computing real eigenvalues of a matrix A for k = 0, 1, 2,... do ρ 0 = 0, x 1 arbitrary (A ρ k I)x k = x k 1, ρ k+1 = x T k Ax k/x T k x k Convergence ρ k λ to an eigenvalue λ of A is cubic if A T = A and quadratic otherwise (in general). Generalized Rayleigh quotient iteration (involving bilinear forms) always converges cubically. ( ) Positive matrices (Frobenius theory)
12 David Hilbert Ostrowski p. 7/3
13 Ostrowski p. 8/3 Algebraic Equations (1933) Critique and correction of Gauss s first and fourth proof of the Fundamental Theorem of Algebra (1939) Continuity of the roots of an algebraic equation as a function of the coefficients: quantitative statement
14 Algebraic Equations (1933) Critique and correction of Gauss s first and fourth proof of the Fundamental Theorem of Algebra (1939) Continuity of the roots of an algebraic equation as a function of the coefficients: quantitative statement Theorem Let x ν, y ν be the zeros of resp. p(z) = a 0 z n + a 1 z n a n, a 0 a n 0 q(z) = b 0 z n + b 1 z n b n, b 0 b n 0. If b ν a ν = ε ν a ν, ε ν ε, 16nε 1/n 1, then, with x ν and y ν suitably ordered, x ν y ν x ν 15nε1/n. Ostrowski p. 8/3
15 Ostrowski p. 9/3 Algebraic Equations (continued) (1940) Long memoir on Graeffe s method ( ) Sign rules (Descartes, Budan-Fourier, Runge) (1979!) A little mathematical jewel
16 Ostrowski p. 9/3 Algebraic Equations (continued) (1940) Long memoir on Graeffe s method ( ) Sign rules (Descartes, Budan-Fourier, Runge) (1979!) A little mathematical jewel Theorem Let p and q be polynomials of degrees m and n, respectively. Define Then M f = max z =1 f(z). γm p M q M pq M p M q, γ = sin m π 8m sinn π 8n.
17 Ostrowski, the skater Ostrowski p. 10/3
18 Ostrowski p. 11/3 Vol. 2. Multivariate Algebra, Formal Algebra Multivariate Algebra ( ) Existence of a finite basis B for a system S of polynomials in several variables. Example: necessary and sufficient conditions for S = {x α 1,i 1 x α 2,i 2 x α n,i n : α ν,i N 0, i = 1, 2, 3,... }
19 Ostrowski p. 11/3 Vol. 2. Multivariate Algebra, Formal Algebra Multivariate Algebra ( ) Existence of a finite basis B for a system S of polynomials in several variables. Example: necessary and sufficient conditions for S = {x α 1,i 1 x α 2,i 2 x α n,i n : α ν,i N 0, i = 1, 2, 3,... } (1924) Theory of invariants of binary forms of degree n
20 Ostrowski p. 11/3 Vol. 2. Multivariate Algebra, Formal Algebra Multivariate Algebra ( ) Existence of a finite basis B for a system S of polynomials in several variables. Example: necessary and sufficient conditions for S = {x α 1,i 1 x α 2,i 2 x α n,i n : α ν,i N 0, i = 1, 2, 3,... } (1924) Theory of invariants of binary forms of degree n (1922, ) Various questions of irreducibility
21 Ostrowski p. 11/3 Vol. 2. Multivariate Algebra, Formal Algebra Multivariate Algebra ( ) Existence of a finite basis B for a system S of polynomials in several variables. Example: necessary and sufficient conditions for S = {x α 1,i 1 x α 2,i 2 x α n,i n : α ν,i N 0, i = 1, 2, 3,... } (1924) Theory of invariants of binary forms of degree n (1922, ) Various questions of irreducibility Formal Algebra (1913!) Algebra of finite fields (37-page memoir in Ukrainian)
22 Ostrowski p. 12/3 Formal Algebra (continued) ( ) Theory of valuation on a field. (Generalizing the concept of absolute value in the field of real numbers to general fields.) Perfect fields and extensions thereof
23 Ostrowski p. 12/3 Formal Algebra (continued) ( ) Theory of valuation on a field. (Generalizing the concept of absolute value in the field of real numbers to general fields.) Perfect fields and extensions thereof (1934) Arithmetic theory of fields (1936) Structure of polynomial rings
24 Ostrowski p. 12/3 Formal Algebra (continued) ( ) Theory of valuation on a field. (Generalizing the concept of absolute value in the field of real numbers to general fields.) Perfect fields and extensions thereof (1934) Arithmetic theory of fields (1936) Structure of polynomial rings (1954) Evaluation of polynomials p n (x) = a 0 x n + a 1 x n a n 1 x + a n by Horner s rule p 0 = a 0, p ν = xp ν 1 + a ν, ν = 1, 2,..., n Complexity: n additions, n multiplications. Optimal for addition, optimal for multiplication when n 4.
25 Formal Algebra (continued) The year 1954 is generally considered the year of birth of algebraic complexity theory (P. Bürgisser and M. Clausen, 1996). (1961) Metric properties of block matrices A = A 11 A 12 A 1n A 21 A 22 A 2n.. A n1 A n2 A nn., A νµ R ν µ, det A νν 0 Example: Is Hadamard s theorem still valid if is replaced by? Answer: yes, if the associated matrix is an M-matrix, where A 11 A 12 A 1n A 21 A 22 A 2n.. A n1 A n2 A nn B = min x =1 Bx,. B = max x =1 Bx Ostrowski p. 13/3
26 Ostrowski p. 14/3 Formal Algebra (continued) (1961) Convergence of block iterative methods for solving Ax = b, where A is a block matrix
27 Ostrowski p. 14/3 Formal Algebra (continued) (1961) Convergence of block iterative methods for solving Ax = b, where A is a block matrix (1977) Kronecker s elimination theory in the general setting of polynomial ideals
28 Ostrowski in his 40s Ostrowski p. 15/3
29 Ostrowski p. 16/3 Vol. 3. Number Theory, Geometry, Topology, Convergence Number Theory (1919) Polynomials with coefficients in a finite arithmetic field taking on integer values for integer arguments. Existence of a regular basis
30 Ostrowski p. 16/3 Vol. 3. Number Theory, Geometry, Topology, Convergence Number Theory (1919) Polynomials with coefficients in a finite arithmetic field taking on integer values for integer arguments. Existence of a regular basis (1919) Arithmetic theory of algebraic numbers
31 Ostrowski p. 16/3 Vol. 3. Number Theory, Geometry, Topology, Convergence Number Theory (1919) Polynomials with coefficients in a finite arithmetic field taking on integer values for integer arguments. Existence of a regular basis (1919) Arithmetic theory of algebraic numbers ( , ) Diophantine equations and approximations
32 Ostrowski p. 17/3 Geometry (1955) A study of evolutes and evolvents of a plane curve (under weak differentiability assumptions). Application to ovals
33 Ostrowski p. 17/3 Geometry (1955) A study of evolutes and evolvents of a plane curve (under weak differentiability assumptions). Application to ovals (1955) Differential geometry of plane parallel curves
34 Ostrowski p. 17/3 Geometry (1955) A study of evolutes and evolvents of a plane curve (under weak differentiability assumptions). Application to ovals (1955) Differential geometry of plane parallel curves (1956) Necessary and sufficient conditions for two line elements to be connectable by a curve along which the curvature is monotone
35 Ostrowski p. 17/3 Geometry (1955) A study of evolutes and evolvents of a plane curve (under weak differentiability assumptions). Application to ovals (1955) Differential geometry of plane parallel curves (1956) Necessary and sufficient conditions for two line elements to be connectable by a curve along which the curvature is monotone Topology (1933) Existence criterion for a common zero of two real functions continuous inside and on the boundary of a disk
36 Ostrowski p. 17/3 Geometry (1955) A study of evolutes and evolvents of a plane curve (under weak differentiability assumptions). Application to ovals (1955) Differential geometry of plane parallel curves (1956) Necessary and sufficient conditions for two line elements to be connectable by a curve along which the curvature is monotone Topology (1933) Existence criterion for a common zero of two real functions continuous inside and on the boundary of a disk (1935) Topology of oriented line elements
37 Ostrowski in his 50s Ostrowski p. 18/3
38 Ostrowski p. 19/3 Convergence (1930) Infinite products (1 + x ν ) = Φ(x) where x 0 = x, x ν+1 = ϕ(x ν ), ν = 0, 1, 2,... ν=0 Example: Euler s product ϕ(x) = x 2, Φ(x) = (1 x) 1. Problem: find all products which converge in a neighborhood of x = 0, and for which ϕ is rational and Φ algebraic.
39 Ostrowski p. 19/3 Convergence (1930) Infinite products (1 + x ν ) = Φ(x) where x 0 = x, x ν+1 = ϕ(x ν ), ν = 0, 1, 2,... ν=0 Example: Euler s product ϕ(x) = x 2, Φ(x) = (1 x) 1. Problem: find all products which converge in a neighborhood of x = 0, and for which ϕ is rational and Φ algebraic. (1930) Normal power series: a ν z ν with a ν 0, a 2 ν a ν 1 a ν+1 ν= and all coefficients between two positive ones are also positive. Theorem: The product of two normal power series is also normal.
40 Ostrowski p. 20/3 Convergence (continued) (1955) Convergence and divergence criteria of Ermakov for f(x)dx
41 Ostrowski p. 20/3 Convergence (continued) (1955) Convergence and divergence criteria of Ermakov for f(x)dx (1956) Fixed point iteration x ν+1 = f(x ν ), ν = 0, 1, 2,..., where f(x) I if x I. Behavior of {x ν } in case of divergence
42 Ostrowski p. 20/3 Convergence (continued) (1955) Convergence and divergence criteria of Ermakov for f(x)dx (1956) Fixed point iteration x ν+1 = f(x ν ), ν = 0, 1, 2,..., where f(x) I if x I. Behavior of {x ν } in case of divergence (1972) Summation of slowly convergent positive or alternating series
43 Ostrowski p. 21/3 Vol. 4. Real Function Theory, Differential Equations, Differential Transformations Real Function Theory ( ) Strengthening, or simplifying, the proof of many known results from real analysis
44 Ostrowski p. 21/3 Vol. 4. Real Function Theory, Differential Equations, Differential Transformations Real Function Theory ( ) Strengthening, or simplifying, the proof of many known results from real analysis (1943) Conditions of integrability for partial differential equations
45 Ostrowski p. 21/3 Vol. 4. Real Function Theory, Differential Equations, Differential Transformations Real Function Theory ( ) Strengthening, or simplifying, the proof of many known results from real analysis (1943) Conditions of integrability for partial differential equations (1946) Indefinite integrals of elementary functions (Liouville Theory)
46 Ostrowski p. 21/3 Vol. 4. Real Function Theory, Differential Equations, Differential Transformations Real Function Theory ( ) Strengthening, or simplifying, the proof of many known results from real analysis (1943) Conditions of integrability for partial differential equations (1946) Indefinite integrals of elementary functions (Liouville Theory) (1952) Convex functions in the sense of Schur, with applications to spectral properties of Hermitian matrices
47 Ostrowski p. 21/3 Vol. 4. Real Function Theory, Differential Equations, Differential Transformations Real Function Theory ( ) Strengthening, or simplifying, the proof of many known results from real analysis (1943) Conditions of integrability for partial differential equations (1946) Indefinite integrals of elementary functions (Liouville Theory) (1952) Convex functions in the sense of Schur, with applications to spectral properties of Hermitian matrices (1957) Points of attraction and repulsion for fixed point iteration in Euclidean space
48 Ostrowski p. 22/3 Real Function Theory (continued) (1958) Univalence of nonlinear transformations in Euclidean space
49 Ostrowski p. 22/3 Real Function Theory (continued) (1958) Univalence of nonlinear transformations in Euclidean space (1966) Theory of Fourier transforms
50 Ostrowski p. 22/3 Real Function Theory (continued) (1958) Univalence of nonlinear transformations in Euclidean space (1966) Theory of Fourier transforms ( ) Study of the remainder term in the Euler-Maclaurin formula
51 Ostrowski p. 22/3 Real Function Theory (continued) (1958) Univalence of nonlinear transformations in Euclidean space (1966) Theory of Fourier transforms ( ) Study of the remainder term in the Euler-Maclaurin formula (1970) The (frequently cited) Ostrowski Grüss inequality f(x)g(x)dx f(x)dx g(x)dx osc f max [0,1] [0,1] g 0
52 Ostrowski p. 23/3 Real Function Theory (continued) (1975) Asymptotic expansion of integrals containing a large parameter
53 Ostrowski p. 23/3 Real Function Theory (continued) (1975) Asymptotic expansion of integrals containing a large parameter (1976) Generalized Cauchy-Frullani integral 0 f(at) f(bt) t where 1 M(f) = lim x x dt = [M(f) m(f)] ln a b, a > 0, b > 0 x 1 f(t)dt, m(f) = lim x 0 x 1 x f(t) t 2 dt
54 Ostrowski s 60th birthday Ostrowski p. 24/3
55 Ostrowski p. 25/3 Differential Equations (1919) Göttingen thesis: Dirichlet series and algebraic differential equations
56 Ostrowski p. 25/3 Differential Equations (1919) Göttingen thesis: Dirichlet series and algebraic differential equations (1956) Theory of characteristics for first-order partial differential equations
57 Ostrowski p. 25/3 Differential Equations (1919) Göttingen thesis: Dirichlet series and algebraic differential equations (1956) Theory of characteristics for first-order partial differential equations (1961) A decomposition of an ordinary second-order matrix differential operator
58 Ostrowski p. 25/3 Differential Equations (1919) Göttingen thesis: Dirichlet series and algebraic differential equations (1956) Theory of characteristics for first-order partial differential equations (1961) A decomposition of an ordinary second-order matrix differential operator Differential Transformations ( ) Various classes of contact transformations in the sense of S. Lie
59 Ostrowski p. 25/3 Differential Equations (1919) Göttingen thesis: Dirichlet series and algebraic differential equations (1956) Theory of characteristics for first-order partial differential equations (1961) A decomposition of an ordinary second-order matrix differential operator Differential Transformations ( ) Various classes of contact transformations in the sense of S. Lie (1942) Invertible transformations of line elements
60 Ostrowski p. 26/3 Vol. 5. Complex Function Theory ( ) Gap theorems for power series and related phenomena of overconvergence
61 Ostrowski p. 26/3 Vol. 5. Complex Function Theory ( ) Gap theorems for power series and related phenomena of overconvergence ( ) Investigations related to Picard s theorem
62 Ostrowski p. 26/3 Vol. 5. Complex Function Theory ( ) Gap theorems for power series and related phenomena of overconvergence ( ) Investigations related to Picard s theorem (1929) Quasi-analytic functions, the theory of Carleman
63 Ostrowski p. 26/3 Vol. 5. Complex Function Theory ( ) Gap theorems for power series and related phenomena of overconvergence ( ) Investigations related to Picard s theorem (1929) Quasi-analytic functions, the theory of Carleman (1933, 1955) Analytic continuation of power series and Dirichlet series
64 Ostrowski p. 27/3 Vol. 6. Conformal Mapping, Numerical Analysis, Miscellanea Conformal Mapping (1929) Constructive proof of the Riemann Mapping Theorem
65 Ostrowski p. 27/3 Vol. 6. Conformal Mapping, Numerical Analysis, Miscellanea Conformal Mapping (1929) Constructive proof of the Riemann Mapping Theorem ( ) Boundary behavior of conformal maps
66 Ostrowski p. 27/3 Vol. 6. Conformal Mapping, Numerical Analysis, Miscellanea Conformal Mapping (1929) Constructive proof of the Riemann Mapping Theorem ( ) Boundary behavior of conformal maps (1955) Iterative solution of a nonlinear integral equation for the boundary function of a conformal map, application to the conformal map of an ellipse onto the disk
67 Ostrowski p. 27/3 Vol. 6. Conformal Mapping, Numerical Analysis, Miscellanea Conformal Mapping (1929) Constructive proof of the Riemann Mapping Theorem ( ) Boundary behavior of conformal maps (1955) Iterative solution of a nonlinear integral equation for the boundary function of a conformal map, application to the conformal map of an ellipse onto the disk Numerical Analysis ( ) Newton s method for a single equation and a system of two equations: convergence, error estimates, robustness with respect to rounding
68 The Ostrowskis in their home in Montagnola Ostrowski p. 28/3
69 Ostrowski p. 29/3 Numerical Analysis (continued) (1953) Convergence of relaxation methods for linear n n systems, optimal relaxation parameters for n = 2
70 Ostrowski p. 29/3 Numerical Analysis (continued) (1953) Convergence of relaxation methods for linear n n systems, optimal relaxation parameters for n = 2 (1954) Matrices close to a triangular matrix A = [a ij ], a ij m (i > j), a ij M (i < j), 0 < m < M Theorem All eigenvalues of A are contained in the union of disks i D i, D i = {z C : z a ii δ(m, M)}, where δ(m, M) = Mm 1 n mm 1 n M 1 n m 1 n. The constant δ(m, M) is best possible.
71 Ostrowski p. 30/3 Numerical Analysis (continued) (1956) Absolute convergence of iterative methods for solving linear systems
72 Ostrowski p. 30/3 Numerical Analysis (continued) (1956) Absolute convergence of iterative methods for solving linear systems (1956) Convergence of Steffensen s iteration
73 Ostrowski p. 30/3 Numerical Analysis (continued) (1956) Absolute convergence of iterative methods for solving linear systems (1956) Convergence of Steffensen s iteration (1957) Approximate solution of homogeneous systems of linear equations
74 Ostrowski p. 30/3 Numerical Analysis (continued) (1956) Absolute convergence of iterative methods for solving linear systems (1956) Convergence of Steffensen s iteration (1957) Approximate solution of homogeneous systems of linear equations (1958) A device of Gauss for speeding up iterative methods
75 Ostrowski p. 31/3 Numerical Analysis (continued) (1964) Convergence analysis of Muller s method for solving nonlinear equations
76 Ostrowski p. 31/3 Numerical Analysis (continued) (1964) Convergence analysis of Muller s method for solving nonlinear equations (1967) Convergence of the fixed point iteration in a metric space in the presence of rounding errors
77 Ostrowski p. 31/3 Numerical Analysis (continued) (1964) Convergence analysis of Muller s method for solving nonlinear equations (1967) Convergence of the fixed point iteration in a metric space in the presence of rounding errors (1967) Convergence of the method of steepest descent
78 Ostrowski p. 31/3 Numerical Analysis (continued) (1964) Convergence analysis of Muller s method for solving nonlinear equations (1967) Convergence of the fixed point iteration in a metric space in the presence of rounding errors (1967) Convergence of the method of steepest descent (1969) A descent algorithm for the roots of algebraic equations
79 Ostrowski p. 31/3 Numerical Analysis (continued) (1964) Convergence analysis of Muller s method for solving nonlinear equations (1967) Convergence of the fixed point iteration in a metric space in the presence of rounding errors (1967) Convergence of the method of steepest descent (1969) A descent algorithm for the roots of algebraic equations (1971) Newton s method in Banach spaces
80 Ostrowski p. 31/3 Numerical Analysis (continued) (1964) Convergence analysis of Muller s method for solving nonlinear equations (1967) Convergence of the fixed point iteration in a metric space in the presence of rounding errors (1967) Convergence of the method of steepest descent (1969) A descent algorithm for the roots of algebraic equations (1971) Newton s method in Banach spaces ( ) a posteriori error bounds in iterative processes
81 Ostrowski p. 32/3 Today s Special (1964) A determinant with combinatorial numbers ( x ) ( x ) ( k 0 k 0 x ) 1 k 0 m ( x ) ( x ) ( k 1 k 1 x ) 1 k 1 m ( x ) ) ( k m x ) ( x k m 1 k m m
82 Ostrowski p. 32/3 Today s Special (1964) A determinant with combinatorial numbers ( x ) ( x ) ( k 0 k 0 x ) 1 k 0 m ( x ) ( x ) ( k 1 k 1 x ) 1 k 1 m ( x ) ) ( k m x ) ( x k m 1 k m m = [Γ(x + 1)] m+1 m µ=1 (x + µ)m+1 µ (k 0,..., k m ) m µ=0 [Γ(k µ + 1)Γ(x m k µ )] where (k 0,..., k m ) = λ>µ(k λ k µ )
83 Ostrowski in his 90s Ostrowski p. 33/3
84 Ostrowski p. 34/3 Vol. 6. Miscellaneous Miscellaneous papers on probability distribution functions
85 Ostrowski p. 34/3 Vol. 6. Miscellaneous Miscellaneous papers on probability distribution functions Lectures
86 Ostrowski p. 34/3 Vol. 6. Miscellaneous Miscellaneous papers on probability distribution functions Lectures Book Reviews
87 Ostrowski p. 34/3 Vol. 6. Miscellaneous Miscellaneous papers on probability distribution functions Lectures Book Reviews Obituaries (G. H. Hardy, ; W. Süss, ; Werner Gautschi, )
88 Gentilino Ostrowski p. 35/3
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