Hilbert Matrix - Historical Note

Historical Note

Hilbert (1894) introduced the Hilbert matrix to study the following question in approximation theory: "Assume that I = is a real interval. Is it then possible to find a non-zero polynomial P with integral coefficients, such that the integral

is smaller than any given bound ε > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the determinant of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length ba of the interval is smaller than 4.

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