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In 1925Centuries: 19th century 20th century 21st century Decades: 1870s 1880s 1890s 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s Years: 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 See also 1925 in aviation 1925 in film 1925 in literature 1925 in mu he started work in the area of almost periodic functionIn mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (is, simplifying the approach of Harald BohrHarald Bohr ( 22 April 1887 22 January 1951) was a Danish mathematician, and younger brother of the physicist Niels Bohr. Many friends remarked that the two men were unusually close, even for brothers. Early in the lives of the brothers, Harald was though by use of compactness and approximate identityIn functional analysis, a right approximate identity in a Banach algebra A is a net (or a sequence) : such that for every element of , the net (or sequence) : has limit. Similarly, a left approximate identity is a net : such that for every element of , th arguments. In 1933 he defined the Bochner integral , as it is now called, for vector-valued functions. Bochner's theorem on Fourier transforms appeared in a 1932 book. His techniques came into their own as Pontryagin duality and then the representation theory of locally compact group s were development in the following years.

Subsequently he worked on multiple Fourier series , posing the question of the Bochner-Riesz mean s. This led to results on how the Fourier transform on Euclidean space behaves under rotations.

In differential geometry, Bochner's formula on curvature from 1946 was most influential. Joint work with Kentaro Yano (1912-1993) led to the 1953 book Curvature and Betti Numbers. It had broad consequences, for the Kodaira vanishing theory , representation theory, and spin manifold s. Bochner also worked on several complex variables (the Bochner-Martinelli formula and the book Several Complex Variables from 1948 with W. T. Martin ).