It arises in the least squares approximation of arbitrary functions by polynomials.
Is it then possible to find a non-zero polynomial P with integer coefficients, such that the integral is smaller than any given bound ε > 0, taken arbitrarily small?"
He concludes that the answer to his question is positive if the length b − a of the interval is smaller than 4.
The Hilbert matrix is symmetric and positive definite.
The determinant of the n × n Hilbert matrix is where Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence OEIS: A005249 in the OEIS), which also follows from the identity Using Stirling's approximation of the factorial, one can establish the following asymptotic result: where an converges to the constant
[1] It follows that the entries of the inverse matrix are all integers, and that the signs form a checkerboard pattern, being positive on the principal diagonal.
For example, The condition number of the n × n Hilbert matrix grows as
The method of moments applied to polynomial distributions results in a Hankel matrix, which in the special case of approximating a probability distribution on the interval [0, 1] results in a Hilbert matrix.
This matrix needs to be inverted to obtain the weight parameters of the polynomial distribution approximation.