Asymptotics of singular numbers of smooth kernels via trigonometric transforms

Inequalities related to smoothness conditions and Fourier series

“…Leindler [17], Theorem 1, (4)). See also Oehring [24], p. 58. In Remark 3 (i), we mentioned that Theorem 3 is a partial generalization of Askey's result [1].…”

On Grasia's criterion for uniform convergence of Fourier series

“…However, by Remark 2.1 and the triangle inequality slowly varying functions are sums of products of iterated logarithms. See [22] and [16] …”

“…(A propos, if the small o is modified to O in the Dini-Lipschitz condition there exists a function that satisfies the relaxed hypothesis but possesses a Fourier series that diverges on a set dense in [0, 27i]; see [22, p. 303 [22, p. 188] or [16]). We now employ 3.3, 3.4, interchange of the order of summation, and (a), (b), and (c) to conclude that…”

“…For example, 3.6, applied to the function W 0 (t) = J^ L (3 k . Observe that (a) condensation is valid for sums of slowly varying terms, (b) E " L(k) = 0{nL(n)), and (c) L(n) = o(T{n)) (see [22, p. 188] or [16]). We now employ 3.3, 3.4, interchange of the order of summation, and (a), (b), and (c) to conclude that…”

Exact asymptotic behavior of singular values of a class of integral operators

“…The asymptotic properties of the spectrum of operators with a convolution kernel have been considered in many papers [1]- [6], [8]- [11], [14], [15]. The exact asymptotics have been obtained under the assumption that the Fourier transform of the kernel satisfies some conditions concerning the rate of growth.…”