By Tarantello G.
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Additional info for A quantization property for blow up solutions of singular Liouville-type equations
0/. y/ j D1 is the reproducing kernel of Fz1;2 . Therefore the sequence fÁj g forms a basis of Fz1;2 . per 1 1 The eigenvalues j of W1 D APP1 APP1 W Fz1;2 ! 2 k/2 for k D 1; 2; : : : . 1 We now turn to the space F1;2 of non-periodic functions. 2 kx/ for x 2 Œ0; 1. 2 k/2 / 1 It is easy to check that g belongs to F1;2 and is orthogonal to all Áj . 0/ D 2 , hence g … F1;2 . x/: 1 Then hf 2 Fz1;2 . 0/ g C hf 1 for all f 2 F1;2 . f /g. f /2 kgk2F 1 C khf k2F 1 : 1;2 1;2 1;2 1 F1;2 Hence, approximation of functions from the unit ball of with n information evaluations is not harder than approximation of periodic functions from the unit ball of 1 Fz1;2 with n 1 information evaluations, and not easier than the periodic case with non-per n evaluations.
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. Strong polynomial tractability holds iff we can take s D 0 in the formula above.
For any m, and for all f 2 F we have kf kG Ä kf kF . Let Pd;m denote the linear space of polynomials of d variables which are of degree at most m in each variable. m C 1/d and kf kF D kf kG for all f 2 Pd;m . m C 1/d -dimensional subspace. As we shall see this property will be very important for our analysis. For the classes Fd;p and Gd;m;p , we consider the multivariate approximation problem APPd with APPd W Fd;p ! Gd;m;p given by APPd f D f: 24 3 Twelve Examples This is clearly a well-defined problem.
A quantization property for blow up solutions of singular Liouville-type equations by Tarantello G.