This volume is a selection from the 281 published papers of Joseph Leonard Walsh, former US Naval Officer and professor at University of Maryland and Harvard University. The nine broad sections are ordered following the evolution of his work. Commentaries and discussions of subsequent development are appended to most of the sections. Also included is one of Walsh's most influential works, "A closed set of normal orthogonal function," which introduced what is now known as "Walsh Functions".
About the Author
Joseph Leonard Walsh (1895-1973) was an American mathematician. For most of his professional career he studied and worked at Harvard University. He received a B.S. in 1916 and a PhD in 1920. The Advisor of his PhD was Maxime Bôcher. He started to work as lecturer in Harvard afterwards and became a full professor in 1935. With two different scholarships he was able to study in Paris under Paul Montel (1920-21) and in Munich under Constantin Carathéodory (1925-26). From 1937 to 1942 he served as chairman of his department at Harvard. During World War II he served as an officer in the US navy and was promoted to captain right after end of the war. After his retirement from Harvard in 1966 he accepted a position at the University of Maryland where he continued to work up to a few months before his death. Joseph L. Walsh became a member of the National Academy of Sciences in 1936 and served 1949-51 as president of the American Mathematical Society. Altogether he published 279 articles (research and others), seven books and advised 31 PhD students. The Walsh function and the Walsh–Hadamard code are named after him. The Grace-Walsh-Szegő Coincidence Theorem is important in the study of the location of the zeros of multivariate polynomials.
Table of Contents1 Zeros and Critical Points of Polynomials and Rational Functions.- [18*] On the location of the roots of the Jacobian of two binary forms and of the derivative of a rational function.- [21-a*] On the location of the roots of the derivative of a polynomial.- [21-b*] On the location of the roots of the Jacobian of two binary forms and of the derivative of a rational function.- [20-c*] On the location of the roots of the derivative of a polynomial.- [33-1*] Note on the location of the roots of the derivative of a polynomial.- [22-g*] On the location of the roots of certain types of polynomials.- [64-e*] A Theorem of Grace on the zeros of polynomials, revisited.- [64-j*] The location of the zeros of the derivative of a rational function, revisited.- [24-h*] An inequality for the roots of an algebraic equation.- Commentary.- Comments on [18*], [21-a*], and [21-b*].- Comments on [20-c*] and [33-i*].- Comments on [22-g*].- Editors’ Note.- Comments on [64-e*], [64-j*].- Comments on [24-h*].- 2 Walsh Functions.- [23-b*] A closed set of normal orthogonal functions.- Commentary.- The Walsh System.- The Impact of Walsh Functions on Modern Mathematics.- Probability Theory.- Harmonic Analysis.- Functional Analysis.- Generalizations.- Technical Applications.- Commentary by T. J. Rivlin.- 3 Qualitative Approximation.- [26-b*] Über die Entwicklung einer analytischen Funktion nach Polynomen.- [26-c*] Über die Entwicklung einer Funktion einer komiexen Veränderlichen nach Polynomen.- [28-a*] On the expansion of analytic functions in series of polynomials and in series of polynomials and in series of other analytic functions.- [28-d*] Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen.- [29-b*] The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions.- Commentary.- 4 Conformai Mapping.- [37-d*] On the shape of level curves of Green’s function.- [38-a*] Note on the curvature of orthogonal trajectories of level curves of Green’s functions.- [39-d*] On the circles of curvature of the images of circles under a conformal map.- [40-a*] Note on the curvature of the orthogonal trajectories of level curves of Green’s functions.- [70-a*] On the shape of the level loci of harmonic measure.- [55-a*] (With D. Gaier) Zur Methode der variablen Gebiete bei der Randverserrung.- [56-b*] (With L. Rosenfeld) On the boundary behavior of a conformal map.- [56-d*] On the conformal mapping of multiply connected regions.- Commentary Dieter Gaier.- Topic I: Geometry of level curves and related topics.- I.1. Domains convex in one direction.- I.2. Length and area problems.- I.3. On the geometry of lemniscates.- Topic II: Conformal mapping near the boundary.- II.1. Conformal mapping of strip domains.- II.2. Hölder continuity of the mapping function.- Topic III: Conformal mapping of multiply connected domains.- III. 1. Walsh’s new canonical map.- III. 2. New approaches to Walsh’s theorem.- III. 3. General canonical domains.- 5 Polynomial Approximation.- [32-c*] On polynomial interpolation to analytic functions with singularities.- [37-g*] (With W.E. Sewell) Note on the relation between continuity and degree of polynomial approximation in the complex domain.- [38-d*] (With W.E. Sewell) Note on degree of trigonometric and polynomial approximation to an analytic function.- [53-c*] (With T.S. Motzkin) On the derivative of a polynomial and Chebyshev approximation.- [73-a*] (With T.S. Motzkin) Equilibrium of inverse-distance forces in three-dimensions.- [34-c*] Note on the orthogonality of Tchebycheff polynomials on confocal ellipses.- [42-a*] Note on the coefficients of overconvergent power series.- [51-c*] Note on approximation by bounded analytic functions.- [68-e*] Approximation by bounded analytic functions: Uniform convergence as implied by mean convergence.- [73-c*] History of the Riemann mapping theorem.- Commentary.- Comments on [32-c*].- Comments on [37-g*] and [38-d*].- Comments on [53-c*] and [73-a*].- Comments on [34-c*].- Comments on [42-a*].- Comments on [51-c*].- Comments on [73-c*].- 6 Rational Approximation 465.- [31-c*] On the overconvergence of certain sequences of rational functions of best approximation.- [34-b*] On approximation to an analytic function by rational functions of best approximation.- [40-b*] On the degree of convergence of sequences of rational functions.- [46-c*] Overconvergence, degree of convergence, and zeros of sequences of analytic functions.- [64-a*] Padé approximants as limits of rational functions of best approximation.- [65-i*] The convergence of sequences of rational functions of best approximation with some free poles.- [67-c*] An extension of the generalized Bernstein lemma.- [68-a*] Degree of approximation by rational functions and polynomials.- [71-b*] (With Dov Aharonov) Some examples in degree of approximation by rational functions.- Commentary.- Comments on [31-c*].- Comments on [34-b*].- Comments on [40-b*].- Comments on [46-c*].- Comments on [64-a*].- Comments on [65-i*].- Comments on [67-c*].- Comments on [68-a*].- Comments on [71-b*].- 7 Spline Functions.- [65-b*] (With J.H. Ahlberg and E.N. Nilson) Fundamental properties of generalized splines.- [67-d*] (With J.H. Ahlberg and E.N. Nilson) Complex cubic splines.- [68-f*] (With J.H. Ahlberg and E.N. Nilson) Cubic splines on the real line.- Commentary by Walter Schempp.