# Walter Rudin Research Materials

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Walter Rudin
Born May 2, 1921
Vienna, Austria
Died May 20, 2010 (aged 89)
Nationality American
Fields Mathematics
Institutions Professor Emeritus, University of Wisconsin-Madison
Alma mater Duke University (B.A. 1947, Ph.D. 1949)
Doctoral advisor John Jay Gergen
Doctoral students Charles Dunkl
Known for Mathematics textbooks; contributions to harmonic analysis and complex analysis[1]
Notable awards American Mathematical Society Leroy P. Steele Prize for Mathematical Exposition

Walter Rudin (May 2, 1921 – May 20, 2010)[2] was an American mathematician and professor of Mathematics at the University of Wisconsin–Madison.

He is known for three books on mathematical analysis: Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis. The first (affectionately referred to as "Baby Rudin") was written when Rudin was a Moore instructor at MIT for his undergraduate analysis course and is widely used as a textbook for undergraduate courses in analysis.

## Biography

Rudin was born into a Jewish family in Austria in 1921. They fled to France after the Anschluss in 1938. When France surrendered to Germany in 1940, Rudin fled to England and served in the British navy for the rest of the war. After the war he left for the United States, and earned his B.A. from Duke University in North Carolina in 1947, and two years later earned a Ph.D. from the same institution. After that he was a C.L.E. Moore instructor at MIT, briefly taught in the University of Rochester, before becoming a professor at the University of Wisconsin–Madison. He remained at the University for 32 years.[2] His research interests spanned from harmonic analysis to complex analysis. He received an honorary degree from the University of Vienna in 2006. His Erdős number is 2.

In 1953, he married fellow mathematician Mary Ellen Estill. The two resided in Madison, Wisconsin, in the eponymous Walter Rudin House, a home designed by architect Frank Lloyd Wright. They had four children.[1]

Rudin died on May 20, 2010 after suffering from Parkinson's disease.[2]

## Publications

### Uniqueness Theory for Laplace Series [3]

Rudin wrote his dissertation while working towards his PhD under doctoral advisor John Jay Gergen at Duke University. It was published in March 1950.

Graphic showing papers cited in Rudin's dissertation and papers citing his dissertation. Arrows direct from original paper to the one citing it.[4]

#### Introduction

In this article, Rudin aims to determine under which conditions a given series of spherical surface harmonics is a Laplace series. Uniqueness properties have been studied for a number of orthogonal systems in one variable, with finite ranges of orthogonality, however this paper is one of the first in which the uniqueness problem is considered for a system which is orthogonal over a two-dimensional set.

A Laplace series is a solution to the Laplace equation. This solution is defined as follows:

$f(r, \theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, r^\ell \, Y_\ell^m (\theta, \varphi ),$

where the $f_\ell^m$ are constants and the factors $r^\ell \, Y_\ell^m$ are known as solid harmonics and $(r,\theta,\varphi)$ are spherical coordinates.

#### Important Definitions

Definition I: The function $Y_n(P)$ defined on a set, S, is said to be a spherical surface harmonic of degree n if: $H_n(x,y,z)=H_n(r,P)=r^nY_n(P)$ is a homogeneous harmonic polynomial in $(x,y,z)$.

Definition II: We say that the series $\sum_{n=0}^\infty Y_n(P)$ of a spherical surface harmonics is of class K if the series $\sum_{n=1}^\infty Y_n(P)/n(n+1)$ is the Laplace series of a function continuous on S.

Definition III: Suppose the series $\sum_{n=0}^\infty Y_n(P)$ is of class K. Let $F(P)$ be the continuous function whose Laplace series is $\sum_{n=1}^\infty Y_n(P)/n(n+1)$.

Definition IV: Given the series $\sum_{n=0}^\infty Y_n(P)$, define the functions $f^*(P)= \limsup_{r\to 1} \sum_{n=0}^\infty Y_n(P)r^n$ and $f_*(P)= \liminf_{r\to 1} \sum_{n=0}^\infty Y_n(P)r^n$

Definition V: A closed set Z on S is said to be of capacity zero on S if Z is a proper subset of S and if the stereographic image of Z in a tangent plane, with center of projection in S - Z, is a plane set of capacity zero.

#### Statement of Main Results

The main results of Rudin's dissertation are summarized in two theorems and a few corollaries, given below:

Theorem I: Given a series of spherical surface harmonics, $\sum_{n=0}^\infty Y_n(P)$, let Z be a closed set of capacity zero on S and suppose:

(i) the given series is of class K

(ii) $\psi^*F(P)$ and $\psi_*F(P)$ are finite on S-Z where $F(P)$ is the continuous function whose Laplace series is given by $\sum_{n=1}^\infty Y_n(P)/n(n+1)$

(iii) there exists a function $y(P)$, defined on S and y in L in S such that $y(P) for P on S
Then the given series is Riemann summable almost everywhere on S and is the Laplace series of its Riemann sum.

Theorem II: Given a series of spherical surface harmonics, $\sum_{n=0}^\infty Y_n(P)$, having $f^*(P)$ and $f_*(P)$ upper and lower Poisson sums, respectively, and let Z be a closed set of capacity zero. Suppose
(i) the given series is of class K
(ii) $f^*(P)$ and $f_*(P)$ are finite on S-Z
(iii) there exists a function $y(P)$ defined on S, $y\in L$ on S, such that $y(P) for P on S
Then the given series is Poisson summable almost everywhere on S and is the Laplace series of its Poisson sum.

Corollary I: If the series $\sum_{n=0}^\infty Y_n(P)$ is of class K and is summable to zero on S, except possibly on a closed set of capacity zero, then the series vanishes identically.

Corollary II: If the two series, $\sum_{n=0}^\infty Y_n(P)$ and $\sum_{n=0}^\infty Y'_n(P)$ are of class K and if they are summable to the same function $f(P)$ (where it is not necessary that $f\in L$ on S) except possibly on a closed set of capacity zero, then the two series are identical.

## Major awards

### Books

• Principles of Mathematical Analysis
• Real and Complex Analysis[5]
• Functional Analysis
• Fourier Analysis on Groups[6]
• Function Theory in Polydiscs
• Function Theory in the Unit Ball of Cn[7]
• The Way I Remember It (autobiography, 1991)

## References

1. ^ a b
2. ^ a b c Ziff, Deborah (May 21, 2010). "Noted UW-Madison mathematician Rudin dies at 89". Wisconsin State Journal. Retrieved May 21, 2010.
3. ^ Rudin, Walter (1950). Uniqueness Theory for Laplace Series (Ph.D.). Duke University.
4. ^ "Web of Science". 4/8/13.
5. ^ Victor L. Shapiro (1968). "Review: Walter Rudin, Real and complex analysis". Bull. Am. Math. Soc. 74 (1): 79–83.
6. ^ J.-P. Kahane (1964). "Review: Walter Rudin, Fourier analysis on groups". Bull. Am. Math. Soc. 70 (2): 230–232.
7. ^ Steven G. Krantz (1981). "Review: Walter Rudin, Function theory in the unit ball of Cn". Bull. Am. Math. Soc. (N. S.) 5 (3): 331–339.
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