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 Royal Navy for the rest of World War II, after which he left for the United States. He obtained both his B.A. in 1947 and Ph.D. in 1949 from Duke University. After his Ph.D., he was a C.L.E. Moore instructor at MIT. He briefly taught at the University of Rochester before becoming a professor at the University of Wisconsin–Madison where he remained for 32 years. His research interests ranged from harmonic analysis to complex analysis.