By Novak E., Wozniakowski H.
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J /, as claimed. Œ0; 1d /. j / d;i d;j ıi;j D Sd Ád; ;i ; Sd Ád; ;j d ˝ ˝ ˛ ˛ D Ád; ;i ; Sd Sd Ád; ;j d; D Ád; ;i ; Wd Ád; ;j d; : Thus, Wd Ád; ;j is orthogonal to all Ád; Wd Ád; D ;j ;i with i 6D j , and therefore for all j 2 Nd . j / D u for all j D Œj1 ; j2 ; : : : ; jd for which jk 2 only for all k 2 u. Recall that Qd d;j D kD1 jk with the ordered j . Œ0;1d / D p max juj d;u 2 : uÂf1;2;:::;d g This allows us to compute the sums of powers of the eigenvalues as Md; . / D d;; C P1 P ;6DuÂf1;2;:::;d g 2 j D2 j d;u maxuÂf1;2;:::;d g Á juj 1= 2juj d;u 2 : Hence, polynomial tractability holds iff sup d D1;2;::: d s C d;; P ;6DuÂf1;2;:::;d g P1 d;u maxuÂf1;2;:::;d g 2 j D2 j 2juj d;u 2 Á juj 1= < 1 for some s 0 and > 0.
We have polynomial intractability, which obviously implies polynomial intractability for the original non-periodic case. But we have weak tractability for the periodic case, and it is not yet clear whether this good property extends to the non-periodic case. x; 0/. 1 Tractability in the Worst Case Setting 33 Weak tractability for m D 0 and p D 2 We now show that weak tractability holds not only for the original non-periodic case but it also holds for a much larger space of less smooth functions.
Y/ if jxi yi j 2 f0; 1g for all i . That is, the values of all derivatives are the same if a component xi D 0 of x is changed into xi D 1. Hence, let per Fd;2 D f f 2 Fd;2 j f is periodicg: per The space Fd;2 is equipped with the same norm as Fd;2 . 2 jk x/ per belong to Fd;2 . L2 D 1. per d;2 The subspace Fd;2 is much smaller than Fd;2 . So if we establish a negative result per for Fd;2 , then the same result will be also true for the larger class Fd;2 . Obviously, per positive results for Fd;2 do not have to be true for Fd;2 .