Elias M. Stein, a mathematician who pioneered work in a field that was originally devised to describe the vibrations of strings but that proved to have a wide range of applications, including charting the fluctuations of stock markets and gravitational waves, died on Dec. 23 at a hospital in Somerville, N.J. He was 87.

The cause was complications of lymphoma, his daughter, Karen, said.

Dr. Stein “had a knack of asking the right question,” said Terence Tao, a professor of mathematics at the University of California, Los Angeles, who was a graduate student of his. “He had this great vision of where mathematics should go.”

Dr. Stein’s specialty was harmonic analysis, which breaks mathematical functions into a spectrum of simple components. It has been known for a long time that a musical note, for example, is a combination of pure harmonic tones. The basic techniques of harmonic analysis, devised centuries ago, were used to compute phenomena like the orbits of planets, the vibrating of strings and the properties of radio signals.

Dr. Stein showed how a similar approach could be applied to other problems and reveal hidden structures and patterns.