Curvature Element Transformations Which Preserve Integrable Fields.

I Shohat, J., Bull. Am. Math. Soc., 41, 51 (1935). 2 Haar, A., Math. Ann., 78, 121-136 (1918). 3 Young, W. H., Comptes Rendus, 165, 696-699 (1917). 4Szego, G., Math. Zeit., 12, 61-94 (1922). b A general treatment, which sets forth necessary and sufficient conditions for equivalence of expansions in terms of orthogonal functions, is given by Walsh, J. L., and Wiener, N., Jour. Math. and Phys. of the Mass. Inst. Tech., 1, 103-122 (1922). 6 For theorems on the order of approximation of continuous functions by polynomials, see Jackson, D., "The Theory of Approximation," Am. Math. Soc. Colloquium, 11, Chap. I. 7The results on the boundedness of orthogonal polynomials to be obtained here are sufficient for the present needs. For a more complete, but less simple, treatment, see Bernstein, S., Jour. Math. pures et appliquees (9), 9, 127-177 (1930), and 10, 219-286 (1931). 8 See Jackson, D., Ann. Math., 34, 533 (1933), Theorem I. 9 Cf. Jackson, loc. cit.,8 p. 534, where further references are given.