For a convex (concave) function f(x):[0,1]|->[0,1].f(0)=0,f(1)=1.f is continuous and increasing on [0,1]. Then (1/(N+1)) sum_(n=1)^Nf(n/N) should be decreasing (increasing) in N.

Cale Terrell

Cale Terrell

Answered question

2022-10-25

For a convex (concave) function f ( x ) : [ 0 , 1 ] [ 0 , 1 ]. f ( 0 ) = 0 , f ( 1 ) = 1. f is continuous and increasing on [ 0 , 1 ]. Then

Answer & Explanation

pawia6g

pawia6g

Beginner2022-10-26Added 14 answers

Assume that f is convex and nondecreasing. Then there exists some nonnegative function g such that f ( x ) = 0 x g ( y ) d y for every x in [ 0 , 1 ] hence the function f is the pointwise limit of some linear combinations with positive coefficients of the functions f z : x ( z x 1 ) + . Let
R N ( f ) = 1 N + 1 n = 1 N f ( n N ) .
Each R N ( f ) is a linear functional of the function f involved hence it is sufficient to solve the case of the functions f z .
The case f 2 is allright (this requires to compute R 2 N ( f 2 ) = R 2 N + 1 ( f 2 ) = N + 1 2 ( 2 N + 1 ) .
Maybe check f 3 ...

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