BULLETIN OF THE 
AMERICAN MATHEMATICAL SOCIETY 
Volume 81, Number 1, January 1975 

JOSEPH L. WALSH 
IN MEMORIAM 

BY MORRIS MARDEN 

Threescore years and ten is the biblical measure of a man's normal 
life span. Yet in modern times his surpassing of this bound is not unusual. 
However, it becomes noteworthy when he is blessed with undiminished 
physical and mental vigor. This was true of Joe Walsh. "His spirit was 
marvelous until the end, and he spoke with gratitude for his many pro-
ductive years."1 

Joseph Leonard Walsh died on December 10, 1973 at the age of seventy 
eight years in his home at University Park, Maryland. This site is not far 
from where he was born on September 21, 1895, as son of Reverend and 
Mrs. John Leonard Walsh. Most of his academic life as student, teacher 
and scholar was spent at Harvard University. In 1916 Harvard awarded 
him the S.B. degree, summa cum laude, and at the same time a Sheldon 
Traveling Fellowship for study at the Universities of Chicago and Wis-
consin. On his return to Harvard in 1917, Walsh began some studies 
under Maxime Bôcher, but their progress was interrupted by World War I 
and his enlistment as an ensign in the U.S. Navy. In 1920 Harvard granted 
him a Ph.D. and also a second Sheldon Traveling Fellowship, this time 
for study in Paris under Paul Montel. Back from Europe in 1921 he joined 
the Harvard faculty, but in 1925 took a leave-of-absence for a year's 
research at Munich under Carathéodory. Returning again to Harvard, 
he was promoted through the ranks to a full professorship in 1935 and 
served as department chairman from 1937 to 1942. In the latter year he 
was recalled to active duty in the U.S. Navy as a lieutenant commander. 
When he returned to Harvard in 1946, he was appointed to the pres-
tigous Perkins Professorship, which he held until his retirement in 1966. 
A semester earlier he began a research professorship at the University of 
Maryland, in which position he remained fully active, working with Ph.D. 
and post doctoral students, until a few months before his death. 

During Walsh's lifetime he received many academic and nonacademic 
honors. Among them was his election in 1936 to the National Academy 
of Science, and in 1937 to the vice-presidency of the American Mathemati-
cal Society. He was elected for a two-year term as president of the Society 
in 1949. This was a crucial period for the Society when it was experiencing 

1 Letter from Mrs. Joseph L. Walsh. 
Copyright © American Mathematical Society 1975 

45 
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46 MORRIS MARDEN [January 

JOSEPH L. WALSH (1895-1973) 

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1975] J. L. WALSH 47 

growing pains due to a rapid increase in membership and in research 
publications. It was the period in which the Society created the post of 
Executive Director and moved its headquarters to Providence. Also 
during this period Walsh served as chairman of the organizing committee 
for the International Mathematical Congress, held in Cambridge during 
August 1950, the first since World War II. At about this time Walsh was 
recipient of a nonmathematical honor—promotion to the rank of captain 
in the U.S. Naval Reserve. In his later years Walsh was twice honored by 
the dedication of volumes of mathematical journals: SIAM J. (2) 3 
(1966) on his seventieth birthday in 1965, and J. Approximation Theory 
5 (1972) on his seventy-fifth birthday in 1970. 

These mathematical honors were well deserved in view of the quantity 
and quality of his original research. Starting with his first publication in 
1916, while still an undergraduate, he wrote, singly or jointly with students 
and others, a total of 279 research, expository, and book review articles 
as well as seven books. Though these papers covered a wide range 
of topics, they were, broadly speaking, concerned with four general 
areas: 

(I) The relative location of the zeros of pairs of rational functions 
such as a polynomial and its derivative. 

(II) Zeros and topology of extremal polynomials. 
(III) The critical points and level lines of Green's function and other 

harmonic functions. 
(IV) Interpolation and approximation of continuous, analytic, or 

harmonic functions. 
Regarding the general area (I), this was Walsh's first main research 

interest. His doctoral thesis was entitled On the roots of the jacobian, of 
two binary forms. It was written under the guidance of Maxime Bôcher 
who had proved that if F and G are binary forms of the same degree and 
if all the zeros aó of F lie in a circular region A and all the zeros bi of G 
lie in a circular region B with BnA = 0, then all the zeros ck of the 
jacobian J(F, G) lie in A UB. Like Bôcher, Walsh used geometric and 
physical methods, interpreting the ck as equilibrium points in the field 
due to positive masses at the points aj and negative masses at the points b$ 
with an inverse distance force law. Walsh's results are generalizations of the 
Lucas theorem that the convex hull of the zeros of a polynomial ƒ contains 
all the critical points of/. These results are described in Walsh's papers and 
in M. Marden, Geometry of polynomials, 2nd éd., Math Surveys, no. 3, 
Amer. Math. Soc, Providence, R.I., 1966, MR 37 #1562. The most 
striking of these results are the following three: 

(1) If an «th degree polynomial ƒ has nx zeros in a disk \z—c1\^.r1 
and the remaining n2=n—n1 zeros in the disk \z—c2|^r2, then any critical 

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48 MORRIS MARDEN [January 

point of/ not in one of these disks lies in a third, "average" disk 

(2) If Cl9 C2 and C3 are disjoint circular regions and if a rational 
function ƒ has in the extended plane all its zeros in CxuC2 and all its 
poles in C3, then any critical point off not in C ^ U Q u C , lies in a circular 
region C4. The boundary 9C4 of C4 is the locus of the point z4 defined by 
the cross ratio (zl9 z2, z3, z4)=const when zl9 z2, z3 vary independently 
on the circles dCl9 dC2, dC3 respectively. 

(3) Let the form 0(zl 5 z2, • • • , zw) be of degree one in each zi9 and of 
total degree n and symmetric in the set zl9 z2, • • • , zn. Let C be a circular 
region containing the n points z,=zi0, y = l , 2, • • • , w. Then in C there 
exists at least one point £ such that <!>(£, £, • • • , Ç)=$(z10, z20, • • • , zn0). 

Regarding the general area (II), the methods and results were suggested 
in part by those in area (I). Given a closed bounded set E containing at 
least n+\ points and the class Pn of all polynomials z

n+a1z
n~1+- • -+an9 

an infrapolynomial p on E means a polynomial p ePn with the property 

max \p(z)\ = min max \q(z)\. 
zeE «ePn zeE 

The zeros of p play a role vis-à-vis set E similar to that of the critical 
points of a polynomial ƒ vis-à-vis the zeros of ƒ. For instance, Fejér 
proved that the zeros ofp lie in the convex hull of E9 and Fekete showed 
that p satisfies a form involving the points of E that is similar to the form 
for the logarithmic derivative of ƒ in terms of the zeros of p. Walsh ex-
plored the subject of infrapolynomials intensively in papers which he 
published singly or jointly with Fekete, Motzkin, Shisha and Zedek. 

Likewise the general area (III) was partly an offshoot of area (I). For 
example, Walsh proved that if G is the Green's function (with pole at 
infinity) for an unbounded region R with bounded boundary B9 then all 
the critical points of G in R lie in the convex hull of B. Thus, the critical 
points of G play a role similar to the critical points of a polynomial whose 
zeros lie on B. In this connection Walsh also examined in detail the cur-
vature and other characteristics of the level lines of Green's function. 
These theorems together with their generalizations to harmonic measures 
and other harmonic functions are developed in Walsh's papers and de-
scribed in his book, Critical points of analytical and harmonic functions. 

As for general area (IV), the subjects of interpolation and approximation 
encompass about half of Walsh's published articles as well as his now 
classical treatise entitled Interpolation and approximation. Among his 
many original results in this area, probably the most important are the 

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1975] J. L. WALSH 49 

following two : 
(1) Every function continuous on a bounded Jordan arc J can be ap-

proximated on J uniformly by a polynomial in z. 
(2) Every function ƒ analytic in a Jordan region B and continuous on 

its closure B can be uniformly approximated on B by a polynomial in z. 
The first is a generalization of Weierstrass' theorem, which requires arc 

J to be a closed interval of the real axis. The second is a generalization of 
Runge's theorem, which requires ƒ to be analytic in B. To prove this 
second theorem, Walsh approximated to B by a sequence of Jordan 
regions Bn with 5<= Bn+1<^Bn for all n, and then applied Runge's theorem 
in B to the function Fn(z)=f(xn(z))9 where w=Xn(

z) maps Bn one-to-one 
conformally onto B. Walsh later extended this second theorem to sets B 
which are the union of a finite number of disjoint Jordan regions. Thus he 
paved the way for the more comprehensive theorem proved later by 
Mergelyan: i f / i s a function continuous on any closed bounded set S 
and analytic at all interior points of S9 then it can be approximated on 
S uniformly by a polynomial in z. 

Walsh maintained an active interest in interpolation, approximation and 
related topics over a period of about fifty years. Most recently he helped 
develop some of the fundamental theorems concerning spline interpolation 
and approximation both on the real line and in the complex plane. 
His contributions may be found in his published papers as well as in the 
monograph The theory of splines and their applications, which he wrote 
jointly with J. H. Ahlberg and E. N. Nilson. 

In the related area of orthogonal expansions, special mention should 
be made of the so-called Walsh functions <p(nXx). These are defined on 
the interval O ^ x ^ l by the relations (p0(x)=l ; 

<p?\x) = l, 0 ^ x < i ; ^(1
1,(x) = - l , * < J C ^ 1 ; 

çCr 1 ^*) = 9>SS(*) = <Pn\2x), 0^x<i; 
tâï^ix) = - fâfa) - ( - 1)*-Vnfc)(2* - 1), i < x <, 1 ; 

A: = 1,2,3,---,2 W - 1 ; « = 1, 2, 3, • • • . 

These functions, having some similarity to the Haar functions, were in-
vented by Walsh in 1923 (see his paper 1923b). Now among the most 
widely used complete orthonormal systems, these functions serve as a 
valuable mathematical tool in communications engineering and other 
applied sciences.2 They are treated in Harmuth's book on Transmission 
of information by orthogonal functions, 2nd éd., Springer-Verlag, New 
York, 1972; in a survey article by Balashov and Rubinshtein, Series with 

2 Letter from T. J. Rivlin. 

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50 MORRIS MARDEN [January 

respect to Walsh system and their generalization, J. Soviet Math. 1 (1973), 
727-763 which lists 140 references; in an article by N. J. Fine, Encyclo-
paedic Dictionary of Physics, suppl. vol. 4, Pergamon, New York, 1971; 
and in Proc. Sympos. on Applications of Walsh Functions, 1970, 1971, 
1972 and 1973. A fifth such symposium was held March 18-20, 1974 
at the Catholic University of America in Washington, D.C. 

Not only did Walsh himself contribute to the above described areas of 
research, but also he introduced many of his students to these areas. 
This is evident from the titles of the doctoral theses written under his direc-
tion. 

Walsh had altogether thirty-one Ph.D. students. The present writer 
succeeded in contacting nearly all of them to ask for their recollections 
of him as a research advisor, teacher and friend. In what follows are 
recorded some of this memorabilia. 

Nearly all his former students agreed that, as a thesis advisor, Walsh 
was patient, generous and considerate, and freely gave help and encourage-
ment. This was true, not only during the developmental stages of each 
thesis, but even during the years afterwards. It must have given him a 
great deal of satisfaction that over the years so many of his students con-
tinued to contribute to mathematical literature, and that they wrote special 
papers in his honor on the occasion of his seventieth and seventy-fifth 
birthdays. 

Their remembrance of Walsh as a teacher almost always includes 
certain rituals accompanying each of his lectures. The opening ritual was 
to fling the classroom windows wide open regardless of outside tempera-
tures and to deliver his lecture pacing back and forth on the platform while 
the students might be freezing in their seats. The closing ritual seemed to 
have been to toss his chalk into the waste basket from whatever position 
he ended the lecture. Besides, during the run of a lecture he was occasion-
ally known to have used an inattentive student as a target for the chalk. 
Elementary calculus and introductory complex function theory were his 
favorite courses, but whatever the subject he prepared his lectures with 
meticulous care and presented them in a deep, musical voice. 

Nearly all his former students were impressed by his love of walking. 
Regardless of the weather he would hike the mile and a half (or so) from 
his home near Fresh Pond to Widener Library or Seaver Hall. When 
he accepted the appointment to the University of Maryland, his first 
prerequisite for a new home was that it be within walking distance of the 
University. 

Some of his former students recollect his sense of humor and his en-
joyment of practical jokes. For example, at the 1948 summer meetings of 
the Society in New Haven, a group photograph was being taken with the 

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1975] J. L. WALSH 51 

mathematicians seated on wide circular bleachers. As the camera rotated, 
Walsh got his companion to run with him from one end of the bleachers 
to the other and thus they appeared twice in the same composite photo-
graph.3 As another instance, a former student who felt himself a novice 
in teaching sought pedagogic advice from Walsh, to which the latter gave 
the terse reply: "Always start writing in the upper left-hand corner of 
the blackboard".4-5 

In summary, Walsh may be characterized as having had a strong sense 
of duty to religion, country and chosen work, as well as a love of art and 
music and a Thoreau-like love of nature.6 Above all, he was a resolute, 
hard worker, known to have spent long hours daily in his Widener study. 
He explored his problems thoroughly along all possible paths and by-
ways, usually reporting his discoveries in a succession of papers. Often 
years later he revisited the same problems, searching for nuggets that he 
may have missed on his earlier explorations. When asked on one occasion 
how he managed to keep up the terrific pace of publication, Walsh 
pointed to the self-portrait of an artist showing Death standing nearby 
and the artist working to complete as much as possible in the little time 
that remained.3 The verdict of history will surely be that Walsh did win 
this race against time. He did leave a permanent imprint upon the mathe-
matics of this century and especially upon the men and women whose 
mathematical careers he helped to launch. 

In the words of the poet, Henry Wadsworth Longfellow, [Charles Sumner, 
stanza 9] 

"So when a great man dies 
For years beyond our ken 

The light he leaves behind him lies 
Upon the paths of men." 

PAPERS BY JOSEPH L. WALSH 

1916 Note on Cauchy's integral formula, Ann. of Math. 18, 79-80. 
1918 On the location of the roots of the Jacobian of two binary forms and of the deriva-

tive of a rational function, Trans. Amer. Math. Soc. 19, 291-298. 
1920 a. On the proof of Cauchy's integral formula by means of Green's formula, Bull. 

Amer. Math. Soc. 26, 155-157. 

3 Letter from E. N. Nilson. 
4 Letter from T. J. Rivlin. 
5 Other anecdotes may be found in the two articles: D. V. Widder, Joseph Leonard 

Walsh, J. SIAM Numer. Anal. 3 (1966), 171-172; M. Marden, Homage to Walsh, 
J. Approximation Theory 5 (1972), ix-xiii. 

6 Letter from Maurice Heins. 

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52 MORRIS MARDEN [January 

b. On the solution of linear equations in infinitely many variables by successive 
approximations, Amer. J. Math. 42, 91-96. 

c. On the location of the roots of the derivative of a polynomial, Ann. of Math. 
22, 128-144. 

1921 a. On the location of the roots of the derivative of a polynomial, Comptes Rendus 
du Congrès International des Mathématiciens (Strasbourg, 1920), Publiés 
par Henri Villat, Toulouse, Edouard Privât, pp. 339-342. 

b. On the location of the roots of the Jacobian of two binary forms and of the deriva-
tive of a rational function, Trans. Amer. Math. Soc. 22, 101-116. 

c. A generalization of the Fourier cosine series, Trans. Amer. Math. Soc. 22, 
230-239. 

d. On the transformation of convex point sets, Ann. of Math. 22, 262-266. 
e. Sur la position des racines des dérivées d'un polynôme, C. R. Acad. Sci. Paris 

172, 662-664. 
f. Solution to proposed problem by Nathan Altschillercourt, "Find the surfaces all 

the plane sections of which are circles", Amer. Math. Monthly 28, 480. 
g. (With N. Wiener), The equivalence of expansions in terms of orthogonal functions, 

J. Math. Phys. 1, 103-122 
h. A theorem on cross-ratios in the geometry of inversion, Ann. of Math. 23, 

45-51. 
1922 a. A theorem on loci connected with cross-ratios, Rend. Circ. Mat. Palermo 

46, 236-248. 
b. On the location of the roots of the derivative of a polynomial, Proc. Nat. Acad. 

Sci. U.S.A. 8, 139-141. 
c. A certain two-dimensional locus, Amer. Math. Monthly 29, 112-114. 
d. Some two-dimensional loci connected with cross ratios, Trans. Amer. Math. 

Soc. 23, 67-88. 
e. A generalization of normal congruences of circles, Bull. Amer. Math. Soc. 

28, 456-462. 
f. On the convergence of the Sturm-Liouville series, Ann. of Math. 24, 109-120. 
g. On the location of the roots of certain types of polynomials, Trans. Amer. Math. 

Soc. 24, 163-180. 
h. On the location of the roots of the Jacobian of two binary forms and of the deriva-

tive of a rational function, Trans. Amer. Math. Soc. 24, 31-69. 
1923 a. Sur un théorème d'algèbre, C. R. Acad. Sci. Paris 176, 1361-1364. 

b. A closed set of normal orthogonal functions, Amer. J. Math. 45, 5-24. 
c. A property of Haar's system of orthogonal functions, Math. Ann. 90, 38-45. 

1924 a. Sur la détermination d'une analytique par ses valeurs sur un contour, C. R. 
Acad. Sci. Paris 178, 58-60. 

b. On the location of the roots of polynomials, Bull. Amer. Math. Soc. 30, 51-62. 
c. Some two-dimensional loci, Quart. J. Pure Math. 1, 154-165. 
d. On the expansion of analytic functions in series of polynomials, Trans. Amer. 

Math. Soc. 26, 155-170. 
e. On the location of the roots of Lamé's polynomials, Tôhoku Math. J. 23, 

312-317. 
f. A generalization of evolutes, Rend. Circ. Mat. Palermo 48, 23-27. 
g. (Book Review), The Strasbourg Congress, Bull. Amer. Math. Soc. 30, 461-464. 
h. An inequality for the roots of an algebraic equation, Ann. of Math. 25, 285-286. 
i. On Pellet's theorem concerning the roots of a polynomial, Ann. of Math. 26, 

59-64. 

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1975] J. L. WALSH 53 

1925 a. Sur la position des racines des fonctions entières de genre zero et un, C. R. Acad. 
Sci. Paris 180,2009-2011. 

1926 a. Note on the location of the roots of a polynomial, Math. Z. 24, 733-742. 
b. Über die Entwicklung einer analytischen Funktion nach Polynomen, Math. Ann. 

96, 430-436. 
c. Über die Entwicklung einer Funktion einer komplexen Verânderlichen nach 

Polynomen, Math. Ann. 96, 437-450. 
d. Über den Gr ad der Approximation einer analytischen Funktion, Sitzungs-

berichte Baverischen Akad. Wiss., 223-229. 
1927 a. A paradox resulting from integeration by parts, Amer. Math. Monthly 34, 88. 

b. On the expansion of harmonie functions in terms of harmonie polynomials, 
Proc. Nat. Acad. Sci. U.S.A. 13, 175-180. 

c. On the degree of approximation to a harmonic function, Bull. Amer. Math. Soc. 
33, 591-598. 

1928 a. On the expansion of analytic functions in series of polynomials and in series of 
other analytic functions, Trans. Amer. Math. Soc. 30, 307-332. 

b. (Book Review), The logarithmic potential, By G. C. Evans, Amer. Math. 
Monthly 35, 254-257. 

c. On approximation to an arbitrary function of a complex variable by polynomials, 
Trans. Amer. Math. Soc. 30, 472-482. 

d. Über die Entwicklung einer harmonischen Funktion nach harmonischen Poly-
nomen, J. Reine Angew. Math. 159, 197-209. 

e. On the degree of approximation to an analytic function by means of rational 
functions, Trans. Amer. Math. Soc. 30, 838-847. 

1929 a. Note on the expansion of analytic functions in series of polynomials and in series 
of other analytic functions, Trans. Amer. Math. Soc. 31, 53-57. 

b. The approximation of harmonic functions by harmonic polynomials and by 
harmonic rational functions, Bull. Amer. Math. Soc. 35, 499-544. 

c. On approximation by rational functions to an arbitrary function of a complex 
variable, Trans. Amer. Math. Soc. 31, 477-502. 

d. Boundary values of an analytic function and the Tchebycheff method of approxi-
mation, Proc. Nat. Acad. Sci. U.S.A. 15, 799-802. 

1930 a. On the overconvergence of sequences of polynomials of best approximation, 
Proc. Nat. Acad. Sci. U.S.A. 16, 297. 

b. Boundary values of an analytic f unction and the Tchebycheff method of approxi-
mation, Trans. Amer. Math. Soc. 32, 335-390. 

c. On the overconvergence of sequences of polynomials of best approximation, 
Trans. Amer. Math. Soc. 32, 794-816. 

1931 a. Note on the overconvergence of sequences of polynomials of best approximation, 
Trans. Amer. Math. Soc. 33, 370-388. 

b. The existence of rational functions of best approximation, Trans. Amer. Math. 
Soc. 33, 668-689. 

c. On the overconvergence of certain sequences of rational functions of best approxi-
mation, Acta Math. 57, 411-435. 

1932 a. On interpolation and approximation by rational functions with preassigned 
poles, Trans. Amer. Math. Soc. 34, 22-74. 

b. An expansion of meromorphic functions, Proc. Nat. Acad. Sci. U.S.A. 18, 
165-171. 

c. On polynomial interpolation to analytic functions with singularities, Bull. Amer. 
Math. Soc. 38, 289-294. 

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54 MORRIS MARDEN [January 

d. On the overconvergence of sequences of rational functions, Amer. J. Math. 
54, 559-570. 

e. On interpolation to harmonic functions by harmonic polynomials, Proc. Nat. 
Acad. Sci. U.S.A. 18, 514-517. 

f. Interpolation and functions analytic interior to the unit circle, Trans. Amer. Math. 
Soc. 34, 523-556. 

1933 a. Bateman on mathematical physics, Bull. Amer. Math. Soc. 39, 178-180. 
(Review). 

b. Interpolation and an analogue of the Laurent development, Proc. Nat. Acad. 
Sci. U.S.A. 19, 203-207. 

c. The Cauchy-Goursat theorem for rectifiable Jordan curves, Proc. Nat. Acad. 
Sci. U.S.A. 19, 540-541. 

d. An extremal problem in analytic functions, Proc. Nat. Acad. Sci. U.S.A. 19, 
900-902. 

e. Note on polynomial interpolation to analytic functions, Proc. Nat. Acad. Sci. 
U.S.A. 19, 959-963. 

f. Note on the location of the critical points of Green* s function, Bull. Amer. Math. 
Soc. 39, 775-782. 

g. A duality in interpolation to analytic functions by rational functions, Proc. Nat. 
Acad. Sci. U.S.A. 19, 1049-1053. 

h. On series of interpolation and the degree of convergence of sequences of analytic 
functions, Tôhoku Math. J. 38, 375-389. 

i. Note on the location of the roots of the derivative of a polynomial, Mathematica 
(Cluj) 8, 185-190. 

1934 a. (With Helen G. Russell), On the convergence and overconvergence of sequences 
of polynomials of best simultaneous approximation to several functions analytic 
in distinct regions, Trans. Amer. Math. Soc. 36, 13-28. 

b. On approximation to an analytic function by rational functions of best approxi-
mation, Math. Z. 38, 163-176. 

c. Note on the orthogonality of Tchebycheff polynomials on confocal ellipses, 
Bull. Amer. Math. Soc. 40, 84-88. 

d. Some interpolation series, Amer. Math. Monthly 41, 300-308. 
e. Sur Vinterpolation par fonctions rationnelles, C. R. Acad. Sci. Paris 198, 1377-

1378. 
f. Note on the location of the critical points of harmonic functions, Proc. Nat. 

Acad. Sci. U.S.A. 20, 551-554. 
1935 a. Lemniscates and equipotential curves of Green's function, Amer. Math. Monthly 

42, 1-17. 
1936 a. A necessary condition for approximation by rational functions, Bull. Amer. 

Math. Soc. 42, 219-221. 
b. The divergence of sequences of polynomials interpolating in roots of unity, 

Bull. Amer. Math. Soc. 42, 715-719. 
c. Note on the behavior of a polynomial at infinity, Amer. Math. Monthly 43, 

461-464. 
d. A mean-value theorem for polynomials and harmonic polynomials, Bull. Amer. 

Math. Soc. 42, 923-930. 
1937 a. Note on the curvature of level curves of Green's function, Proc. Nat. Acad. Sci. 

U.S.A. 23, 84-89. 
b. (With W. E. Sewell), Note on degree of approximation to an integral by Riemann 

sums, Amer. Math. Monthly 44, 155-160. 

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1975] J. L. WALSH 55 

c. Note on the curvature of orthogonal trajectories of level curves of Green's function 
Proc. Nat. Acad. Sci. U.S.A. 23, 166-169. 

d. On the shape of level curves of Green's function, Amer. Math. Monthly 44, 
202-213. 

e. Maximal convergence of sequences of harmonic polynomials, Ann. of Math. 
38, 321-354. 

f. (With G. M. Merriman), Note on the simultaneous orthogonality of harmonic 
polynomials on several curves, Duke Math. J. 3, 279-288. 

g. (With W. E. Sewell), Note on the relation between continuity and degree of 
polynomial approximation in the complex domain, Bull. Amer. Math. Soc. 
43, 557-563. 

1938 a. Note on the curvature of orthogonal trajectories of level curves of Green's 
functions, Bull. Amer. Math. Soc. 44, 520-523. 

b. (With W. Seidel), On the derivatives of functions analytic in the unit circle, 
Proc. Nat. Acad. Sci. U.S.A. 24, 337-340. 

c. On interpolation and approximation by functions analytic and bounded in a 
given region, Proc. Nat. Acad. Sci. U.S.A. 24, 477-486. 

d. (With W. E. Sewell), Note on degree of trigonometric and polynomial approxi-
mation to an analytic function, Bull. Amer. Math. Soc. 44, 865-873. 

1939 a. (Book Review), Convergence, by W. L. Ferrar, Science 89, 59-60. 
b. Note on the location of zeros of the derivative of a rational function whose 

zeros and poles are symmetric in a circle, Bull. Amer. Math. Soc. 45, 462-
470. 

c. On interpolation by functions analytic and bounded in agiven region, Trans. Amer. 
Math. Soc. 46, 46-65. MR 1, 10. 

d. On the circles of curvature of the images of circles under a conformai map, Amer. 
Math. Monthly 46, 472-485. MR 1, 111. 

1940 a. Note on the curvature of orthogonal trajectories of level curves of Green's 
function. Ill, Bull. Amer. Math. Soc. 46, 101-108. MR 1, 210. 

b. On the degree of convergence of sequences of rational f unctions, Trans. Amer. 
Math. Soc. 47, 254-292. MR 1, 309. 

c. Note on the degree of convergence of sequences of analytic functions, Trans. 
Amer. Math. Soc. 47, 293-304. MR 1, 310. 

d. (With W. E. Sewell), Note on degree of trigonometric and polynomial approxi-
mation to an analytic function, in the sense of least pth powers, Bull. Amer. 
Math. Soc. 46, 312-319. MR 1, 309. 

e. (With W. E. Sewell), Sufficient conditions for various degrees of approximation 
by polynomials, Duke Math. J. 6, 658-705. MR 2, 80. 

1941 a. (Book Review), Sur les valeurs exceptionelles des fonctions méromorphes et de 
leurs dérivées, By Georges Valiron, Actualités Sci. Indust., no. 570, Hermann, 
Paris, 1937; Bull. Amer. Math. Soc. 47, 7-8. 

b. (With W. E. Sewell), On the degree of polynomial approximation to analytic 
functions: problem P, Trans. Amer. Math. Soc. 49, 229-257. MR 2, 276. 

c. (With W. Seidel), On approximation by euclidean and non-euclidean trans-
lations of an analytic function, Bull. Amer. Math. Soc. 47, 916-920. MR 4, 10. 

1942 a. Note on the coefficients of over convergent power series, Bull. Amer. Math. Soc. 
48, 163-166. MR 3, 201. 

b. (With W. Seidel), On the derivatives of functions analytic in the unit circle and 
their radii ofunivalence and of p-valence, Trans. Amer. Math. Soc. 52,128-216. 
MR 4, 215. 

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56 MORRIS MARDEN [January 

1943 a. Local civil time and date from time diagram, Proc. U.S. Naval Institute 69 
(Whole No. 479), 23-24. 

b. A new diagram for a universal small-area plotting sheet, Proc. U.S. Naval 
Institute 69 (Whole No. 487), 1221-1222. 

1944 a. (With E. N. Nilson), Interpolation and approximation by functions analytic 
and bounded in a given region, Trans. Amer. Math. Soc. 55,53-67. MR 5,115. 

1946 a. The running fix as used at sea (Some navigational wrinkles on "Pilot Chart of 
the North Atlantic Ocean," No. 1400), Hydrographie Office, Washington, D.C. 

b. Note on the location of the critical points of harmonic functions, Bull. Amer. 
Math. Soc. 52, 346-347. MR 7, 382. 

c. Overconvergence, degree of convergence, and zeros of sequences of analytic func-
tions, Duke Math. J. 13, 195-234. MR 8, 201. 

d. On degree of approximation on a Jordan curve to a function analytic interior to 
the curve by functions not necessarily analytic interior to the curve, Bull. Amer. 
Math. Soc. 52, 449-453. MR 7, 514. 

e. Taylor's series and approximation to analytic f unctions, Bull. Amer. Math. Soc. 
52, 572-579. MR 8, 144. 

f. (Book Review), Table of arc sin x and Tables of associated Legendre functions, 
by Lyman J. Briggs, et al., Science 104, 41. 

g. Note on the location of the zeros of the derivative o f a rational function having 
prescribed symmetry, Proc. Nat. Acad. Sci. U.S.A. 32, 235-237. MR 8, 144. 

1947 a. A rigorous treatment of the first maximum problem in the calculus, Amer. Math. 
Monthly 54, 35-36. 

b. On the location of the critical points of harmonic measure, Proc. Nat. Acad. Sci. 
U.S.A. 33, 18-20. MR 8, 461. 

c. (With E. N. Nilson), Note on the degree of convergence of sequences of poly-
nomials, Bull. Amer. Math. Soc. 53, 116-117. MR 9, 23. 

d. Note on the derivatives of functions analytic in the unit circle, Bull. Amer. Math. 
Soc. 53, 515-523. MR 9, 23. 

e. Note on the critical points harmonic functions, Proc. Nat. Acad. Sci. U.S.A. 
33, 54-59. MR 8, 513. 

f. The location of the critical points of simply and doubly periodic functions, Duke 
Math. J. 14, 575-586. MR 9, 180. 

1948 a. On the critical points of functions possessing central symmetry on the sphere, 
Amer. J. Math. 70, 11-21. MR 9, 428. 

b. Note on the location of the critical points of harmonic functions, Bull. Amer. Math. 
Soc. 54, 191-195. MR 9, 432. 

c. The critical points of linear combinations of harmonic functions, Bull. Amer. 
Math. Soc. 54, 196-205. MR 9, 432. 

d. Critical points of harmonic functions as positions of equilibrium in afield of force, 
Proc. Nat. Acad. Sci. U.S.A. 34, 111-119. MR 9, 432. 

e. Methods of symmetry and critical points o f harmonic f unctions, Proc. Nat. Acad. 
Sci. U.S.A. 34, 267-271. MR 9, 585. 

f. On the location of the zeros of the derivatives of a polynomial symmetric in the ori-
gin, Bull. Amer. Math. Soc. 54, 942-945. MR 10, 250. 

1949 a. (With W. E. Sewell and H. M. Elliott), On the degree of convergence of harmonic 
polynomials to harmonic functions, Proc. Nat. Acad. Sci. U.S.A. 35, 59-62. 
MR 10, 374. 

b. (With E. N. Nilson), On functions analytic in a region: Approximation in the 
sense of least pth powers, Trans. Amer. Math. Soc. 65, 239-258. MR 10, 524. 

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1975] J. L. WALSH 57 

c. (With A. S. Galbraith and W. Seidel), On the growth of derivatives of f une-
tions omitting two values, Trans. Amer. Math. Soc. 67, 320-326. MR 11, 
344. 

d. (With W. E. Sewell and H. M. Elliott), On the degree of polynomial approxima-
tion to harmonic and analytic functions, Trans. Amer. Math. Soc. 67, 381-420. 
MR 11, 515. 

e. (With W. E. Sewell), On interpolation to an analytic function in equidistant 
points: Problem p, Bull. Amer. Math. Soc. 55, 1177-1180. MR 11, 344. 

1950 a. On distortion at the boundary of a conformai map, Proc. Nat. Acad. Sci. U.S.A. 
36, 152-156. M R U , 507. 

b. (With H. Margaret Elliott), Polynomial approximation to harmonic and analytic 
functions: generalized continuity conditions, Trans. Amer. Math. Soc. 68, 
183-203. MR 11, 515. 

c. The location of critical points of harmonic f unctions, Leopoldo Fejér et Frederico 
Riesz LXX Annos Natis Dedicatus, Pars B. Acta Sci. Math. Szeged 12, 61-65. 
MR 12, 26. 

d. (With H. G. Russell), On simultaneous interpolation and approximation by 
functions analytic in a given region, Trans. Amer. Math. Soc. 69, 416-439. MR 
12, 813. 

1951 a. On Rouché's theorem and the integral-square measure of approximation, Proc. 
Amer. Math. Soc. 2, 671-681. MR 13, 335. 

b. Note on the location of the critical points of a real rational function, Proc. Amer. 
Math. Soc. 2, 682-685. MR 13, 451. 

c. Note on approximation by bounded analytic functions, Proc. Nat. Acad. Sci. 
U.S.A. 37, 821-826. MR 13, 545. 

1952 a. On Rouché's theorem and the integral-square measure of approximation, Proc. 
Internat. Congress Math., vol. 1, Amer. Math. Soc., Providence, R.I., pp. 405-
406. 

b. (With E. N. Nilson), Note on overconvergence in sequences of analytic functions, 
Proc. Amer. Math. Soc. 3, 442-443. MR 13, 927. 

c. Polynomial expansions of functions defined by Cauchfs integral, J. Math. Pures 
Appl. (9) 31, 221-244. MR 14, 547. 

d. Note on the location of zeros of extremal polynomials in the non-euclidean 
plane, Acad. Serbe Sci. Publ. Inst. Math. 4, 157-160. MR 14, 164. 

e. (With Philip Davis), Interpolation and orthonormal systems, J. Analyse Math. 
2, 1-28. MR 16, 580. 

f. Degree of approximation to functions on a Jordan curve, Trans. Amer. Math. 
Soc. 73, 447-458. MR 14, 630. 

g. (With H. Margaret Elliott), Degree of approximation on a Jordan curve, 
Proc. Nat. Acad. Sci. U.S.A. 38, 1058-1066. MR 14, 741. 

1953 a. An interpolation series expansion for a meromorphic f unction, Trans. Amer. Math. 
Soc. 74, 1-9. MR 14, 741. 

b. On continuity properties of derivatives of sequences of functions, Proc. Amer. 
Math. Soc. 4, 69-75. MR 14, 736; 1278. 

c. (With T. S. Motzkin), On the derivative of a polynomial and Chebyshev approxi-
mation, Proc. Amer. Math. Soc. 4, 76-87. MR 15, 701. 

d. Note on the shape of level curves of Green's function, Duke Math. J. 20,611-615. 
MR 15, 310. 

e. Note on the shape of level curves of Green's function, Amer. Math. Monthly 
60, 671-674. MR 15, 424. 

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58 MORRIS MARDEN [January 

f. (With David Young), On the accuracy of the numerical solution of the Dirichlet 
problem by finite differences, J. Res. Nat. Bur. Standards 51, 343-363. MR 15, 
562. 

1954 a. An interpolation problem for harmonic functions, Amer. J. Math. 76, 259-272. 
MR 16, 588. 

b. (With David Young), On the degree of convergence of solutions of difference 
equations to the solution of the Dirichlet problem, J. Math. Phys. 33, 80-93. 
MR 15, 746. 

c. (With Philip Davis), On representations and extensions of bounded linear 
functionals defined on classes of analytic functions, Trans. Amer. Math. Soc. 
76, 190-206. MR 15, 803. 

d. (With J. P. Evans), Note on the distribution of zeros of extremal polynomials, 
Proc. Nat. Acad. Sci. U.S.A. 40, 332-337. MR 15, 954. 

e. (With J. P. Evans), On approximation by bounded analytic functions, Arch. 
Math. 5, 191-196. MR 15, 947. 

f. Sur Vapproximation par fonctions analytiques bornées, C. R. Acad. Sci. Paris 
239, 1339-1341. MR 16, 811. 

g. Sur la représentation conforme des aires multiplement connexes, C. R. Acad. 
Sci. Paris 239, 1572-1574. MR 16, 581. 

h. Sur la représentation conforme des aires multiplement connexes, C. R. Acad. 
Sci. Paris 239, 1756-1758. MR 16, 811. 

i. (With M. Fekete), On the asymptotic behavior of polynomials with extremal 
properties, and of their zeros, J. Analyse Math. 4, 49-87. MR 17, 354. 

j . Détermination d'une fonction analytique par ses valeurs données dans une in-
finité dénombrable de points, Bull. Soc. Math. Belg., 52-70. MR 17, 601. 

1955 a. (With D. Gaier), Zur Methode der variablen Gebiete bei der Randverzerrung, 
Arch. Math. 6, 77-86. MR 16, 348. 

b. (With T. S. Motzkin), Least pth power polynomials on a real finite point set, 
Trans. Amer. Math. Soc. 78, 67-81. MR 16, 585. 

c. A generalization of Jensen's theorem on the zeros of the derivative of a polynomial, 
Amer. Math. Monthly 62, 91-93. MR 16, 818. 

d. (With J. P. Evans), On interpolation to a given analytic function by analytic 
functions of minimum norm, Trans. Amer. Math. Soc. 79,158-172. MR 16,1011. 

e. Sur r approximation par fonctions rationnelles et par fonctions holomorphes 
bornées, Ann. Mat. Pura Appl. (4) 39, 267-277. MR 17, 1077. 

1956 a. (With Mishael Zedek), On generalized Tchebycheff polynomials, Proc. Nat. 
Acad. Sci. U.S.A. 42, 99-104. MR 17, 730. 

b. (With L. Rosenfeld), On the boundary behavior of a conformai map, Trans. 
Amer. Math. Soc. 81, 49-73. MR 17, 836. 

c. Best-approximation polynomials of given degree, Proc. Sympos. Appl. Math., 
Vol. VI, Numerical Analysis, Santa Monica, Aug. 1953, Amer. Math. S o c , 
Providence, R.I., pp. 213-218. MR 18, 32. 

d. On the conformai mapping of multiply connected regions, Trans. Amer. Math. 
Soc. 82, 128-146. MR 18, 290. 

e. (With T. S. Motzkin), Least pth power polynomials on a finite point set, Trans. 
Amer. Math. Soc. 83, 371-396. MR 18, 479. 

f. (With J. P. Evans), On the location of the zeros of certain orthogonal functions, 
Proc. Amer. Math. Soc. 7, 1085-1090. MR 18, 725. 

g. Note on degree of approximation to analytic functions by rational functions with 
preassignedpoles, Proc. Nat. Acad. Sci. U.S.A. 42, 927-930. MR 18, 569. 

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19751 J. L. WALSH 59 

h. "Birkhoff, George Davidv, article in Encyclopedia Britannica, 
i. (With M. Fekete), On restricted infrapolynomials, J. Analyse Math. 5, 47-76. 

MR 19, 263. 
1957 a. (With T. S. Motzkin), Underpolynomials and infrapolynomials, Illinois J. 

Math. 1, 406-426. MR 19, 643. 
b. (With T. S. Motzkin), Polynomials of best approximation on a real finite point 

set, Proc. Nat. Acad. Sci. U.S.A. 43, 845-846. MR 19, 852. 
c. (With M. Fekete), Asymptotic behavior of restricted extremal polynomials and 

of their zeros, Pacific J. Math. 7, 1037-1064. MR 19, 1045. 
d. (With David Young), Lipschitz conditions for harmonic and discrete harmonic 

functions, J. Math. Phys. 36, 138-150. MR 20, #2532. 
1958 a. On approximation by bounded analytic functions, Trans. Amer. Math. Soc. 

87, 467-484. MR 20 #3298. 
b. A generalization of Faber's polynomials, Math. Ann. 136, 23-33. MR 21 

#725. 
c. Complex numbers and complex variables, Laplace's differential equation, and 

Conformai mapping, three articles in the McGraw-Hill Encyclopedia of Science 
and Technology. 

d. Approximation by bounded analytic f unctions, Seminars on Analytic Functions, 
vol. II, pp. 73-87, published by U.S. Air Force, Office of Scientific Research, 
Washington, T>.C. 

e. On infrapolynomials with prescribed constant term, J. Math. Pures Appl. 
(9) 37, 295-316. MR 20 #7098. 

1959 a. On extremal approximations, On Numerical Approximation, Univ. of Wiscon-
sin Press, Madison, Wis., pp. 209-216. MR 21 #421. 

b. (With T. S. Motzkin), Location of zeros of infrapolynomials, Compositio 
Math. 14, 50-70. MR 21 #3539. 

c. Approximation on a line segment by bounded analytic functions: Problem (i, 
Proc. Amer. Math. Soc. 10, 270-272. MR 21 #6443a. 

d. Note on least-square approximation to an analytic function by polynomials, as 
measured by a surface integral, Proc. Amer. Math. Soc. 10, 273-279. MR 23 
#A1047. 

e. Approximation by bounded analytic functions: General configurations, Proc. 
Amer. Math. Soc. 10, 280-285. MR 21 #6443b. 

f. (With T. S. Motzkin), Polynomials of best approximation on a real finite point 
set. I, Trans. Amer. Math. Soc. 91, 231-245. MR 21 #7388. 

g. (With H. G. Russell), Integrated continuity conditions and degree of approxima-
tion by polynomials or by bounded analytic functions, Trans. Amer. Math. Soc. 
92, 355-370. MR 21 #7311. 

h. Note on approximation by bounded analytic functions {Problem a), Math. Z. 
72, 47-52. MR 22 #776. 

i. (With T. S. Motzkin), Polynomials of best approximation on an interval, Proc. 
Nat. Acad. Sci. U.S.A. 45, 1523-1528. MR 22 #9773. 

j . Note on invariance of degree of polynomial and trigonometric approximation 
under change of independent variable, Proc. Nat. Acad. Sci. U.S.A. 45, 1528— 
1533. MR23#A1191. 

k. (With H. J. Landau), On canonical conformai maps of multiply connected regions, 
Trans. Amer. Math. Soc. 93, 81-96. MR 28 #4093. 

1. The analogue for maximally convergent polynomials of JentzscKs theorem, 
Duke Math. J. 26, 605-616. 

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60 MORRIS MARDEN [January 

1960 a. Solution of the Dirichlet problem for the ellipse by interpolating harmonic 
polynomials, J. Math. Mech. 9, 193-196. MR 22 #4891. 

b. On the asymptotic properties of extremal polynomials with prescribed constant 
term, Math. Z. 73, 339-345. MR 22 #2715. 

c. Note on polynomial approximation on a Jordan arc, Proc. Nat. Acad. Sci. 
U.S.A. 46, 981-983. MR 22 #12338. 

d. On degree of approximation by bounded harmonic functions, J. Math. Pures 
Appl. (9) 39, 201-220. MR 27 #1609. 

e. (With T. S. Motzkin), Best approximators within a linear family on an interval, 
Proc. Nat. Acad. Sci. U.S.A. 46, 1225-1233. MR 28 #5279. 

f. Degree of approximation by bounded harmonic functions, Proc. Nat. Acad. Sci. 
U.S.A. 46, 1390-1393. MR 22 #12338b. 

g. Note on degree of approximation by bounded analytic functions: Problem p, 
Trans. Amer. Math. Soc. 96, 246-258. MR 22 #11140. 

1961 a. The circles of curvature of the curves of steepest descent of Green's function, 
Amer. Math. Monthly 68, 323-329. MR 23 #A3272. 

b. (With T. S. Motzkin), Conformai maps of small disks, Proc. Nat. Acad. Sci. 
U.S.A. 47, 1838-1843. MR 26 #307. 

c. (With O. Shisha), The zeros of infrapolynomials with some prescribed coef-
ficients, J. Analyse Math. 9, 111-160. MR 25 #174. 

d. (With J. P. Evans), Approximation by bounded analytic functions to functions 
represented by Dirichlet series, Proc. Amer. Math. Soc. 12, 875-879. MR 
25 #4109. 

e. A new generalization of Jensen's theorem on the zeros of the derivative of a poly-
nomial, Amer. Math. Monthly 68, 978-983. MR 24 #A2009. 

1962 a. Degree of polynomial approximation to an analytic function as measured by a 
surface integral, Proc. Nat. Acad. Sci. U.S.A. 48, 26-32. MR 24 #A2176. 

b. (With J. H. Ahlberg and E. N. Nilson), Best approximation properties of the 
spline fit, J. Math. Mech. 11, 225-234. MR 25 #738. 

c. Asymptotic properties of polynomials with auxiliary conditions of interpolation, 
Ann. Polon. Math. 12, 17-24. MR 27 #1605. 

d. (With T. S. Motzkin), Polynomials of best approximation on an interval. II, 
Proc. Nat. Acad. Sci. U.S.A. 48, 1533-1537. MR 26 #2788. 

e. On the convexity of the ovals of lemniscates, Studies in Mathematical Analysis 
and Related Topics, Stanford University Press, Stanford, Calif., pp. 419-423. 
MR 27 #1606. 

f. Approximation par les fonctions holomorphes bornées. Problème $', J. Math. 
Pures Appl. (9) 41, 213-232. MR 27 #5914. 

1963 a. (With T. S. Motzkin), Zeros of the error function for Tchebycheff approximation 
in a small region, Proc. London Math. Soc. (3) 13, 90-98. MR 26 #1667. 

b. Restricted infrapolynomials and trigonometric infrapolynomials, Proc. Nat. 
Acad. Sci. U.S.A. 49, 302-304. MR 27 #2613. 

c. A generalization of Fejér's principle concerning the zeros of extremal polynomials, 
Proc. Amer. Math. Soc. 14, 44-51. MR 27 #271. 

d. A sequence of rational functions with application to approximation by bounded 
analytic functions, Duke Math. J. 30,177-189. MR 30 #2155. 

e. (With O. Shisha), The zeros of infrapolynomials with prescribed values at given 
points, Proc. Amer. Math. Soc. 14, 839-844. MR 27 #3785. 

f. Note on the convergence of approximating rational functions of prescribed type, 
Proc. Nat. Acad. Sci. U.S.A. 50, 791-794. MR 28 #400; 28 #1247. 

g. (Book Review), Analytic function theory, by E. Hille, SIAM Rev. 5, 377-378. 

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19751 J. L. WALSH 61 

1964 a. Padé approximants as limits of rational functions of best approximation, J. 
Math. Mech. 13, 305-312. MR 28 #4283. 

b. (With O. Shisha), On the location of the zeros of some infrapolynomials with 
prescribed coefficients, Pacific J. Math. 14, 1103-1109. MR 30 #232. 

c. (With O. Shisha), Extremal polynomials and the zeros of the derivative of a 
rational function, Proc. Amer. Math. Soc. 15, 753-758. MR 29 #4876. 

d. The convergence of sequences of rational functions of best approximation, Math. 
Ann. 155, 252-264. MR 29 #1484. 

e. A theorem of Grace on the zeros of polynomials, revisited, Proc. Amer. Math. 
Soc. 15, 354-360. MR 28 #4092. 

f. (With Z. Rubinstein), On the location of the zeros of a polynomial whose center 
of gravity is given, J. Analyse Math. 12, 129-142. MR 29 #4877. 

g. (With A. Sharma), Least squares and interpolation in roots of unity, Pacific 
J. Math. 14, 727-730. MR 28 #5278. 

h. Surplus free poles of approximating rational functions, Proc. Nat. Acad. Sci. 
U.S.A. 52, 896-901. MR 30 #3983. 

i. (With Maynard Thompson), Approximation with auxiliary conditions, J. Math. 
Mech. 13, 1015-1019. MR 30 #1253. 

j . The location of the zeros of the derivative of a rational function, revisited, 
J. Math Pures Appl. (9) 43, 353-370. MR 31 #3582. 

1965 a. Geometry of the zeros of the sums of linear fractions, Trans. Amer. Math. 
Soc. 114, 30-39. MR 31 #3579. 

b. (With J. H. Ahlberg and E. N. Nilson), Fundamental properties of generalized 
splines, Proc. Nat. Acad. Sci. U.S.A. 52, 1412-1419. MR 36 #6846. 

c. (With J. H. Ahlberg and E. N. Nilson), Best approximation and convergence 
properties of higher-order spline approximations, J. Math. Mech. 14, 231-243. 
MR 35 #5823. 

d. (With A. Sinclair), On the degree of convergence of extremal polynomials and 
other extremal functions, Trans. Amer. Math. Soc. 115, 145-160. MR 33 
#7564. 

e. The convergence of sequences of rational functions of best approximation. II, 
Trans. Amer. Math. Soc. 116, 227-237. MR 32 #6120. 

f. (With J. H. Ahlberg and E. N. Nilson), Extremal, orthogonality, and conver-
gence properties of multidimensional splines, J. Math. Anal. Appl. 12, 27-48. 
MR 37 #661. 

g. (With J. H. Ahlberg and E. N. Nilson), Convergence properties of generalized 
splines, Proc. Nat. Acad. Sci. U.S.A. 54, 344-350. MR 36 #6847. 

h. Hyperbolic capacity and interpolating rational functions, Duke Math. J. 32, 
369-379. MR 31 #6081. 

i. The convergence of sequences of rational functions of best approximation with 
some free poles, Approximation of Functions (Proc. Sympos. General Motors 
Res. Lab., 1964), Henry L. Garabedian, Ed., Elsevier, Amsterdam, 1965, 
pp. 1-16. MR 32 #4441. 

1966 a. (With T. S. Motzkin), Mean approximation on an interval for an exponent less 
than one, Trans. Amer. Math. Soc. 122, 443-460. MR 34 #1769. 

b. Approximation by polynomials: Uniform convergence as implied by mean con-
vergence, Proc. Nat. Acad. Sci. U.S.A. 55, 20-25. MR 32 #5891. 

c. Approximation by polynomials: Uniform convergence as implied by mean con-
vergence, II. Proc. Nat. Acad. Sci. U.S.A. 55, 1405-1407. MR 35 #4443. 

d. (With H. G. Russell), Hyperbolic capacity and interpolating rational functions. 
II, Duke Math. J. 33, 275-279. MR 33 #1624. 

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62 MORRIS MARDEN [January 

e. The convergence of approximating rational functions of prescribed type, Con-
temporary Problems in the Theory of Analytic Functions (M. A. Lavrent'ev, 
Ed.), Proc. Internat. Conference on the Theory of Analytic Functions 
(Erevan, 1965), "Nauka", Moscow, 1966, pp. 304-308. (Russian). MR 35 
#3069. 

f. Approximation by polynomials: Uniform convergence as implied by mean 
convergence. I l l , Proc. Nat. Acad. Sci. U.S.A. 56, 1406-1408. MR 35 #4444. 

1967 a. Best approximation by rational functions and by meromorphic functions with 
some free poles, J. Analyse Math. 18, 359-375. MR 36 #1673. 

b. On the convergence of sequences of rational functions, SIAM J. Numer. Anal. 
4, 211-221. MR 36 #1675. 

c. An extension of the generalized Bernstein lemma, Colloq. Math. 16, 91-92. 
MR 35 #6844. 

d. (With J. H. Ahlberg and E. N. Nilson), Complex cubic splines, Trans. Amer. 
Math. Soc. 129, 391-413. MR 36 #573. 

1968 a. Degree of approximation by rational functions and polynomials, Michigan Math. 
J. 15, 109-110. MR 36 #6845. 

b. Note on classes of functions defined by integrated Lipschitz conditions, Bull. 
Amer. Math. Soc. 74, 344-346. MR 36 #2807. 

c. (With T. S. Motzkin), A persistent local maximum of the pth power deviation 
on an interval,p<\, Pacific J. Math. 24, 133-142. MR 38 #4868. 

d. The convergence of sequences of rational functions of best approximation. Ill, 
Trans. Amer. Math. Soc. 130, 167-183. MR 36 #1674. 

e. Approximation by bounded analytic functions: Uniform convergence as implied 
by mean convergence, Trans. Amer. Math. Soc. 130, 406-413. MR 36 #3997. 

f. (With J. H. Ahlberg and E. N. Nilson), Cubic splines on the real line, J. Approxi-
mation Theory 1, no. 1, 5-10. MR 37 #6650. 

1969 a. (With E. B. Saff), Extensions of D. Jackson's theorem on best complex poly-
nomial mean approximation, Trans. Amer. Math. Soc. 138, 61-69. MR 39 
#3001. 

b. Inequalities expressing degree of convergence of rational functions, J. Approxi-
mation Theory 2, 160-166. MR 39 #7115. 

c. (With J. H. Ahlberg and E. N. Nilson), Properties of analytic splines. I. Com-
plex poly nominal splines, J. Math. Anal. Appl. 27, 262-278. MR 42 #8136. 

d. Note on approximation by bounded analytic functions, Problem a: General 
configurations, Aequationes Math. 3, 160-164. MR 41 #5632. 

e. Approximations to a function by a polynomial in a given function, Amer. Math. 
Monthly 76, 1049-1050. 

f. (With Z. Rubinstein), Extensions and some applications of the coincidence 
theorems, Trans. Amer. Math. Soc. 146, 413-427. MR 40 #4428. 

1970 a. (With W. J. Schneider), On the shape of the level loci of harmonic measure, 
J. Analyse Math. 23, 441-460. MR 42 #6205. 

b. Approximation by rational f unctions : Open problems, J. Approximation Theory 
3, 236-242. MR 43 #538. 

c. Note on degree of convergence of sequences of rational functions of prescribed 
type, Proc. Nat. Acad. Sci. U.S.A. 67, 1188-1191. MR 42 #4748. 

1971 a. (With J. H. Ahlberg and E. N. Nilson), Complex polynomial splines on the 
unit circle, J. Math. Anal. Appl. 33, 234-257. MR 43 #3696. 

b. (With Dov Aharonov), Some examples in degree of approximation by rational 
functions, Trans. Amer. Math. Soc. 159, 427-444. MR 44 #6974. 

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19751 J. L. WALSH 63 

c. Mean approximation by polynomials on a Jordan curve, J. Approximation 
Theory 4, 263-268. MR 45 #3726. 

1972 a. (With Dov Aharonov), On the convergence of rational functions of best approxi-
mation to a meromorphic function, J. Math. Anal. Appl. 40, 418-426. MR 47 
#5263. 

b. (With Myron Goldstein), Approximation by rational functions on Riemann 
surfaces, Proc. Amer. Math. Soc. 36, 464-466. MR 47 #2072. 

c. Note on the convergence of sequences of rational functions, Proc. Nat. Acad. 
Sci. U.S.A. 69, 2963-2964. MR 46 #7522. 

1973 a. (With T. S. Motzkin), Equilibrium of inverse distance forces in three dimensions, 
Pacific J. Math. 44, 241-250. MR 47 #5234. 

b. (With E. B. Saflf), On the convergence of rational functions which interpolate 
in the roots of unity, Pacific J. Math. 45, 639-641. 

c. History of the Riemann mapping theorem, Amer. Math. Monthly 80, 270-276. 
1974 a. The role of the pole in rational approximation, Rocky Mountain Math. J. 

(to appear), 
b. (With P. M. Gauthier and Alice Roth), Uniform approximation in the spherical 

metric (in preparation). 

BOOKS BY JOSEPH L. WALSH 

1. Approximation by Polynomials in the Complex Domain, Mémorial des Sciences 
Mathématiques, Gauthier-Villars, Paris, 1935, ii+72 pp. 

2. Interpolation and Approximation by Rational Functions in the Complex Domain, 
Colloquium Publications, Vol. 20, American Mathematical Society, Providence, R.I., 
1935, ix-f382pp.; 2nd edition, 1952; 3rd edition, 1960; 4th rev. edition, 1965; 
5th edition, 1969; Russian translation, IL, Moscow, 1961. MR 36 #1671; 1672b, c. 

3. A Bibliography on Orthogonal Polynomials (with J. A. Shohat and Einar Hille), 
National Research Council, Bulletin No. 103, Washington, D.C., 1940, ix+204 pp. 

4. The Location of Critical Points of Analytic and Harmonic Functions, Colloquium 
Publications, Vol. 34, American Mathematical Society, Providence, R.I., 1950, 
viii+384pp. MR 12, 249. 

5. Approximation by Bounded Analytic Functions, Mémorial des Sciences Mathé-
matiques, Fase. 144, Gauthier-Villars, Paris, 1960, 66 pp. MR 22 #9770. 

6. A Rigorous Treatment of Maximum-Minimum Problems in the Calculus, D. C. Heath, 
Boston, Mass., 1962, 22 pp. 

7. The Theory of Splines and Their Applications, (with J. H. Ahlberg and E. N. Nilson), 
Academic Press, New York and London, 1967, xi+284 pp. MR 39 #684. 

WALSH'S P H . D . STUDENTS 
(With journal reference to some or all their thesis results) 

1928 MORRIS MARDEN, On the location of the roots of the jacobian of two binary 
forms and of the derivative of a rational function, Trans. Amer. Math. Soc. 32 
(1930), 81-109. 

1930 ORIN J. FARRELL, On the expansion of harmonic f unctions in series of harmonic 
polynomials belonging to a simply connected region, Amer. J. Math. 57 (1935). 

1931 CECIL T. HOLMES, Approximation of harmonic functions in 3 dimensions by 
harmonic polynomials (not published). 

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64 MORRIS MARDEN [January 

1932 JOSEPH L. DOOB, Boundary values of analytic functions, Trans. Amer. Math. 
Soc. 34 (1932), 153-170; ibid 35 (1933), 418-451; On a theorem of Gross and 
Iversen, Ann. of Math. 33 (1932), 753-757. 

1932 HELEN G. RUSSELL, On the convergence and overconvergence of sequences of 
polynomials of best simultaneous approximation to several functions analytic in 
distinct regions, Trans. Amer. Math. Soc. 36 (1934), 13-28. 

1935 JOHN H. CURTISS, Interpolation in regularly distributed points, Trans. Amer. 
Math. Soc. 38 (1935), 458-473; On the Jacobi series, Trans. Amer. Math. Soc. 
49 (1941), 467-501. MR 2, 355. 

1935 YU-CHENG SHEN, On interpolation and approximation by rational functions 
with preassignedpoles, J. Chinese Math. Soc. 1 (1936), 154-173. 

1936 WALTER E. SEWELL, Generalized derivatives and approximation by polynom-
ials, Trans. Amer. Math. Soc. 41 (1937), 84-120. 

1937 ZEHMAN I. MOSESSON, Maximal sequences of polynomials (not published). 
1938 FLOYD E. ULRICH, Problem of type for a certain class of Riemann surfaces, 

Duke Math. J. 5 (1939), 567-589. MR 1, 8. 
1940 MAURICE H. HEINS, Extremal problems for functions analytic and single-

valued in a doubly-connected region, Amer. J. Math. 62 (1940), 91-106. MR 1, 
114; On the iteration of functions which are analytic and single-valued in a given 
multiply-connected region, Amer. J. Math. 63 (1941), 461-480. MR 2, 275. 

1940 ABRAHAM SPITZBART, Approximation in the sense of least pth powers with a 
single auxiliary condition of interpolation, Bull. Amer. Math. Soc. 52 (1946), 
338-346. MR 7, 425. 

1941 EDWIN N. NILSON, Interpolation and approximation of analytic functions by 
functions analytic and bounded in a given region-, Interpolation and approximation 
by functions analytic and bounded in a given region', (published with J. L. Walsh), 
Trans. Amer. Math. Soc. 55 (1944), 53-67; MR 5, 115; ibid. 65 (1949), 239-
258. MR 10, 524. 

1942 LYNN H. LOOMIS, The radius and modulus of n-valence for analytic functions 
where first n—\ derivatives vanish at a point, Bull. Amer. Math. Soc. 46 (1940), 
496-501. MR 1, 308; On an inequality of Seidel and Walsh, 48 (1942), 908-911. 
MR 5, 37. 

1947 IVAN R. HERSHNER, Jr., Radii ofunivalence andp-valence of functions analytic 
in the unit circle (not published). 

1948 H. MARGARET ELLIOTT, On the degree of approximation to harmonic func-
tions by harmonic polynomials, Trans. Amer. Math. Soc. 67 (1949), 381-420. 

1949 HELLEN KELSALL NICKERSON, Studies in overconvergence, Bull. Amer. 
Math. Soc, 55 (1949), 1061. 

1952 ALAN F. KAY, Distribution of zeros of sequences of polynomials of unbounded 
degree, Proc. Amer. Math. Soc. 6 (1955), 571-582. MR 17, 247. 

1952 ISAAC E. BLOCK, Kernel functions and class L2, Proc. Amer. Math. Soc. 4 
(1953), 110-117. MR 14, 989; Duke Math. J. 19 (1952), 367-378. MR 14, 153. 

1953 THEODORE J. RIVLIN, On sufficient conditions for overconvergence, Proc. 
Amer. Math. Soc. 6 (1955), 597-602. MR 17, 138. 

1953 LAWRENCE ROSENFELD, On the boundary of a conformai map, Trans. 
Amer. Math. Soc. 81 (1956), 49-73. 

1953 JACQUELINE P. EVANS, On approximation and interpolation by functions ana-
lytic in a given region and an application to orthonormal systems, Arch. Math. 5 
(1954), 191-196; Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 332-337; Trans. Amer. 
Math. Soc. 79 (1955), 158-172; Proc. Amer. Math. Soc. 7 (1956), 1085-1090. 

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19751 J. L. WALSH 65 

1954 RICHARD S. VARGA, Semi-infinite and infinite strips of zeros, Univ. e Poli-
tccnio Torino, Rend. Sem. Math. 11 (1951/52), 289-296. MR 14 #546. 

1956 MISHAEL ZEDEK, On generalized Tchebycheff polynomials, Proc. Nat. 
Acad. Sci. U.S.A. 42 (1956), 99-104. 

1957 HENRY J. LANDAU, On canonical conformai maps of multiply connected 
domains, Trans. Amer. Math. Soc. 99 (1961), 1-20. MR 22 #12212. 

1961 VINCENT C. WILLIAMS, On conformai maps of regions of infinite connectivity, 
Trans. Amer. Math. Soc. 155 (1971), 427-453. 

1962 DOROTHY B. SHAFFER, Shape of level loci of Green's function and other 
harmonic functions, J. Analyse Math. 17 (1966), 59-70; J. Math. Mech. 19 
(1969), 41-48. 

1963 VICTOR M. MANJARREZ, Polynomial bases for compact sets in the plane, 
Trans. Amer. Math. Soc. 132 (1968), 541-551. 

1964 JERRY L. FIELDS, Rational approximations to hypergeometric functions, 
Math. Comp. 19 (1965), 606-624. 

1968 EDWARD B. SAFF, Interpolation and functions of class H(k, a, 2), J. Approxi-
mation Theory 1 (1968), 488-492. MR 39 #2992; Trans. Amer. Math. Soc. 
138 (1969), 61-69; ibid 141 (1969), 79-92. MR 39 #4406. 

1973 MARVIN E. ORTEL, Approximation by bounded analytic functions. 

Current address: Department of Mathematics, University of Wisconsin, Milwaukee, 
Wisconsin 53201 

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