Appendix to Frigyes Riesz and Bela Sz. -Nagy Functional by Bela Sz. -Nagy

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But it is known that the sum of the multiplicities of the zeros can exceed the degree of the polynomial only when Q (z) is identically zero. This proves that P. (f; x) will be unique. It is clear that the interpolating polynomial P. (f; x) can be written in the form rm akr-1 P . 2) i=0 where the Lk,i(x) are polynomials of degree

4) m ak-1 Z-x =1 {=0 Lk, i(x) i! (z - xk)41. 4) to be a function of the parameter z. 4) is the expansion in sums of simple fractions. We note that the point z = x is a simple pole of R. (__L_; x) with residue equal to unity. 3. 4) to a common denominator. Setting M A (z) =II(z - xk)ak k=1 we obtain a fractional representation for R. 5) is proper the numerator B (z, x) is a polynomial in z of degree not greater than n + 1. We can show that B (z, x) is inde- pendent of z and equals A W. 5) for values of z with large modulus.

J p (x) P. (x) P, (x) dx = a 0 form # n 1 form = n. We will write the nth degree polynomial of an orthonormal system in the form P. (x) = a"x" + b"xn-1 +.. 9) We now prove that three consecutive polynomials of an orthonormal system satisfy a recursion relation Preliminary Information 22 / - bn+,. - W. + C5,kPk(z). k-0 The coefficients cn,k are the Fourier coefficients: p (x) xPn (x) Pk (x) day Cn,k a If k < n - 1 then zPk (x) is a polynomial of degree k + 1 < n and cn,k = 0 because P. (x) is orthogonal to each polynomial of degree less than n, xPn (x) = Cn,n+1 Pn+1 (x) + cn,n Pn (x) + cn, n-1Pn_1(x).

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