Research Work
Luzin's first significant result was a construction of an almost everywhere divergent trigonometric series with monotonic convergence to zero coefficients (1912). This example disproved the Pierre Fatou conjecture and was unexpected to most mathematicians at that time.
At approximately the same time, he proved what is now called Lusin's theorem in real analysis.
His Ph.D. thesis entitled Integral and trigonometric series (1915) had a large impact on the subsequent development of the metric theory of functions. A set of problems formulated in this thesis for a long time attracted attention from mathematicians. For example, the first problem in the list, on the convergence of the Fourier series for a square-integrable function, was solved by Lennart Carleson in 1966.
In the theory of boundary properties of analytic functions he proved an important result on the invariance of sets of boundary points under conformal mappings (1919).
Luzin was one of the founders of descriptive set theory. Together with his student Mikhail Yakovlevich Suslin, he developed the theory of analytic sets.
He also made contributions to complex analysis, the theory of differential equations, and numerical methods.
Read more about this topic: Nikolai Luzin
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