Iii troduc tioiiThe expansion of smooth functions into trigonometric sine series is descrjbed in a detailed way in JONES/HARDY [GI, SHAW/JOHNSON/RIESS [lo], and TASCHE [ll].If a function x ( t ) is defined on [0, 11 and is sufficiently smooth, then the asymptotic magnitude of its FOCRIER coefficients x : in the expansion(1)(2) and the order of convergence of its associated series expansion (1) depend only on the boundary ralues of x(t,, more explicitly on the largest integer p with 1 6 (3) x(?k)(O)=x(2'~)(1)=0 ( k = O , . . . , p -1) .If x ( t ) is smooth but (3) fails to be satisfied, then a polynomial y ( t ) can be determined which corrects the boundary values of x(t). This polynomial P-1(4 solves the Lidstone interpolation problem, where the Lidstone polynomials Ak(t) in (4) are recursively defined by ( 5 ) Ao(t) =t , A;+l(t) =a&) , Ak+l(O) =&+I( 1) = 0 .TAWHE gives an &-error estimate for the partial sums of the corrected function z ( t ) -y ( t ) . JONES/HARDY state an order of convergence in the TCHEBYCHEV norm.In the sequel we mill prove sharper e-estimates than JONES/HARDY'S (which are
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