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Reade[10] has recently improved Weyl's classical estimate λn = o(n−3/2) for the eigenvalues of a symmetric kernel K∈C1 by relaxing the Cl hypothesis to the assumptions that K∈L2[0, 2π]2, that K is absolutely continuous in each variable separately, and that both ∂K/∂s and ∂K/t belong to L2[0, 2π]2. The conclusion of his theorem, that is, of course, stronger than λn = o(n−3/2).

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

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Garsia's discovery that functions in the periodic Besov space A(p~', p, 1), with 1 < p < oo , have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in A(p~', p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to A(a, p, q).