Let E be a set of positive measure on the unit circle.Let ƒ e H p (1 f£ p ^ oo) and g be the restriction off to JE.It is shown that functions g x , A > 0, can be constructed from g so that g k -• ƒ.We also characterize those functions g on E which are restrictions of functions in H p (1 < p £ oo).In the following, the space H p (1 ^ p ^ oo) will, according to the context, be either the Hardy class of analytic functions in the open unit disc D or the space of the corresponding boundary value functions, viz the subspace of "analytic" functions in L P (C\ C being the unit circle.If E c C has positive measure then it is well known (see [3]) that a function in H p cannot vanish on E without being identically zero.Thus, theoretically at least, ƒ e H p is uniquely "determined" by its values on £.In the present work we address ourselves to the problem of recovering functions in H p from their restrictions to E. Theorem I gives an explicit constructive solution to this problem.The allied problem of characterizing the restrictions to E of functions in H p (1 < p ^ oo) is solved in Theorem II.To the best of our knowledge, the only known results relating to these problems are due to the author [4] where the case p = 2 is dealt with.THEOREM!Let E c C with m(E) > 0. Suppose that 1 ^ p ^ oo, feH p and that g is the restriction of f to E. For each X > 0 define analytic functions h x ,g x onD by h x (z) -exp|-i-log(l + X) f Ç±ldei, zeD, g x {z) = kh x iz)-\ * ' SK zeD.Ml JE W -z Then as k -• oo, g k -• ƒ uniformly on compact subsets of Z).Moreover for 1 < p < oo we also have \\g x -ƒ || p -» 0 as X -+ oo.THEOREM II.Let E c C with 0 < m(£) < m(C).For geL l (E) let g x be as in Theorem I. (a) If I < p < oo then a function g eL p (E) is the restriction to E of some f e H p if and only if sup A > 0 \\g x \ \ p < oo.(b) A function g e L°°(£) is the restriction to E of some f e H 00 if and only ifsup p>1 lim sup^^^Hp < oo.

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