On the Role of the Lebesgue Functions in the Theory of the Lagrange Interpolation'

where for n 1, 2 . . . . we have (1 .2) 1 fix,,,>xz„> . . .>x,,,, Then, as it is well known, for given values y,,,, there is exactly one polynomial g(x) of degree ~ aI such that g(x, •„)(r'1, 2, . . ., n) . If the values y, ., . are the values f(x,,,,) of a function f(x) defined in [-1, 1], then we call the corresponding g(x) polynomial "the nr'l interpolatory polynomial of f(x) belonging to A" and denote it by L,(f, A) or -if misunderstanding cannot arise by L,,(f) . The abscissae x,.,, are called the n ut fundamental points of the matrix A and are sometimes denoted also by .x,, . It is well known that L ;, (f, A) can be written in the form