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St. Petersburg: Academy of Science, 1866. Item #70

1. O funktsiyakh, naimenee uklonyaushchikhsya ot nulya [i.e. On the Functions with the Smallest Deviation from Zero]. St. Petersburg: Academy of Science, 1873. 32 pp. 25x16 cm.
2. O razlozhenii funktsiy v ryady pri pomoshchi nepreryvnykh drobey [i.e. On Functions Decomposition Using Continuous Fractions]. St. Petersburg: Academy of Science, 1866. 26 pp. 25x16 cm. No wrappers as issued. Uncut. Near fine.

Two rare imprints of Pafnuty Lvovich Chebyshev's (1821-1894) important contributions to the theory of approximation of functions. In both works Chebyshev wrote about phenomenon now known as Chebyshev polynomials (a sequence of orthogonal polynomials which can be defined recursively), their qualities and features. Chebyshev polynomial is a polynomial with the largest possible leading coefficient, but subject to the condition that their absolute value on the interval [-1,1] is bounded by 1. Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials, which are also called Chebyshev nodes, are used as nodes in the optimal polynomial interpolation of an arbitrary function. This discovery of Chebyshev lies in the fundament of the modern theory of approximation.
Imprint ‘O funktsiyah’ is important because in it Chebyshov presents the polynomial in the form it’s known now. Chebyshev was called by Russian historians 'the mathematician equally great to Lobachevsky' and best known for his theory of the distribution of the Prime numbers, works on probability theory, the theory of approximation of functions and applied mechanics. He managed to find some fundamental results in these fields – there are function, polynomial, inequality, equation and several theorems named after him.

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