3.19 Nonnegative weighted sums and integrals. (a) Show that f(x) = L;-1 aixji] is a convex function of x, where ai 2 a2 > ·> ar > 0, and æi] denotes the ith largest component of x. (You can use the fact that f(x) = E-1 "[ is convex on R".) (b) Let T(x,w) denote the trigonometric polynomial
3.19 Nonnegative weighted sums and integrals. (a) Show that f(x) = L;-1 aixji] is a convex function of x, where ai 2 a2 > ·> ar > 0, and æi] denotes the ith largest component of x. (You can use the fact that f(x) = E-1 "[ is convex on R".) (b) Let T(x,w) denote the trigonometric polynomial
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please send handwritten solution for 3.19 part a
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![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<w < 27}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fae079-28af-4fb7-b5d6-8bc2e6b2edbf%2F0ce2dc8c-cb37-4815-88dc-ff586c65021f%2Fkxgoonh_processed.jpeg&w=3840&q=75)
Transcribed Image 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<w < 27}.
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