3.19 Nonnegative weighted sums and integrals. (a) Show that f(x) ar > 0, and |2] denotes the ith largest component of x. (You can use the fact that f (x) = E, x1a] is convex on R".) 2- aixji] is a...

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a2 > ·..> (b) Let T(x,w) denote the trigonometric polynomial T(x,w) = x1 + x2 cos w + x3 cos 2w + ... + xn cos(n – 1)w. Show that the function c2n f(x) = - | log T(x,w) dw is convex on {x € R" | T(x,w) > 0, 0< 27}.="" "/="">
Extracted text: 3.19 Nonnegative weighted sums and integrals. (a) Show that f(x) ar > 0, and |2] denotes the ith largest component of x. (You can use the fact that f (x) = E, x1a] is convex on R".) 2- aixji] is a convex function of x, where a1 > a2 > ·..> (b) Let T(x,w) denote the trigonometric polynomial T(x,w) = x1 + x2 cos w + x3 cos 2w + ... + xn cos(n – 1)w. Show that the function c2n f(x) = - | log T(x,w) dw is convex on {x € R" | T(x,w) > 0, 0<>