Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician renowned for his foundational work in series theory, particularly the concept of Cesàro summation, and for his significant contributions to differential geometry.
Born in Naples, Kingdom of the Two Sicilies (now Italy), Cesàro pursued his early studies at the Royal School of Engineers in Liège, Belgium, before earning his doctorate from the University of Rome. He subsequently held academic positions as a professor at the University of Palermo and later at the University of Naples.
Cesàro's most celebrated contribution to mathematics is the method of Cesàro summation, a technique used to assign a sum to certain infinite series that do not converge in the traditional sense. This method involves considering the sequence of arithmetic means of the partial sums of a series. If this sequence of averages converges to a limit, the original series is said to be Cesàro summable to that limit. This concept proved instrumental in fields such as Fourier series analysis, analytic number theory, and the study of divergent series.
Beyond his work on series, Cesàro was a prolific researcher in differential geometry, where he introduced innovative methods and results concerning curves and surfaces. He published numerous articles and authored an influential two-volume textbook, Lezioni di Geometria Intrinseca (Lectures on Intrinsic Geometry), which explored infinitesimal calculus and differential geometry. His intellectual pursuits also extended to areas like number theory and mathematical physics.
Cesàro's rigorous approach and creative solutions left a profound and lasting impact on several branches of mathematics, establishing him as one of the leading Italian mathematicians of his era. He died in Torre Annunziata, Italy, at the age of 47.