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By Gobel R., Hill P., Liebert W. (eds.)

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INTRODUCTION It is the aim of this paper to continue the investigations on the degree of the best appro- ximation of functions by singular integrals with positive kernels. One of the authors first treated these problems with semi-group theory [2, 3] , and later introduced integral transform methods 2 ), in particular Fourier-transfarm methods [ 4, 5, 6], in case the singular integrals are convolution integrals connected with the Fourier-transform. By the latter method one can generalize the results in two respects: The function space under consideration need not be reflexive, a condition very useful in the semi-group approach, and the singular integrals need not be semi-group Operators.

_o). uk(u)äu Proof of the. First assume that the first moment of k (u) exists. Then by lemma 4. 3. we have q1 (o) J q1 (t) dt - J 00 = 0 0 J uk(u)du= 00 00 [1-k 1 (t)}dt= lim tq1 (t)+ ttoo A 0 and furthermore • q1 (s) = s • [1- k(s) -1 l for Res > o. Applying Lemma 4. 1, then (4. 3) follows. Conversely, if the limit of (4. 5 Res > o, it also exists for s = cr > J! q1 (u) du t < oo, we have by Lemma 4. 2 and 4. 3 that the limit is equal o. of k (u) . This completes the proof. For o < 'Y < 1, we have J e- st q'Y (t) dt 00 (Res > o) , 0 where (4.

Furthermore for every c > o and independent of p > o . 13) Jt {f(u) - J 0 p (u)} du (p > o) Approximation and Laplace transforms ('e-ct ldG (t)l = p'Y p 0 35 ('e-ct lf(t)- J (t)l dt = 0(1) p 0 for every c > 0 . This completes the proof. From the point of view of the saturation class of a given singular integral, we may combine these three theorems to give the following Let e - ct f E L 1 (o, oo) for every c > o, and let k satisfy the condition ( P) . 11) is fulfilled, then the integral THEOREM 5.

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