page_title: Charles Jean de la Vallée Poussin
section_title: Work
hierarchical_section_title: Charles Jean de la Vallée Poussin / Work
context_page_description: Charles-Jean Étienne Gustave Nicolas, baron de la Vallée Poussin was a Belgian mathematician. He is best known for proving the prime number theorem.
  The king of Belgium ennobled him with the title of baron.
context_section_description: Although his first mathematical interests were in analysis, he became suddenly famous as he proved the prime number theorem independently of his coeval Jacques Hadamard in 1896.
  Afterwards, he found interest in approximation theory. He defined, for any continuous function f on the standard interval , the sums
  ,
  where
  and
  are the vectors of the dual basis with respect to the basis of Chebyshev polynomials (defined as
  .
  Note that the formula is also valid with  being the Fourier sum of a -periodic function  such that
  .
  Finally, the de la Vallée Poussin sums can be evaluated in terms of the so-called Fejér sums (say )
  .
  The kernel is bounded () and obeys the property
  , if 
  Later, he worked on potential theory and complex analysis.
  He also published a counterexample to Alfred Kempe's false proof of the four color theorem. The Poussin graph, the graph he used for this counterexample, is named after him.
caption: The Poussin graph