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be the orthonormal trigonometric sums on Di for weight p(6)cr(0) ; if the u's and v's are uniformly bounded on a point set D2 contained in Di, the same is true of the U's and Vs. For this case the proof admits a materially simpler formulation than when geometric configurations are contemplated having the degree of generality previously considered. The details relating to the loci C', C", K, K', K.", can be dispensed with for the most part ; with 0 replacing the pair of coordinates (x, y), and $ replacing the pair (u, v), it is sufficient, for any particular value of 0, to consider separately the intervals (0-T/2, 0+T/2) and ( 0 + T T / 2 , 0 + 3 T T / 2 ) , and in the integral corresponding to the right-hand member of (5) to represent Xn_i(0, </>) in the former interval by an expression with denominator sin (0—</>), and in the latter interval by an alternative expression with 1—cos (0— </>) in the denominator.

Approximation of continuous functions by means of lacunary polynomials