Bdd linear functionals on L-inf

• Feb 13th 2011, 01:32 PM
subfallen
Bdd linear functionals on L-inf
Is it possible to find a function g(x) ∈ L1([0,1]) so that:

∫x^n g(x)dx = δ_{n1}

For n = 0, 1, 2,..., N where N is finite? (The right hand side denotes the Kronecker delta.)
• Feb 14th 2011, 04:36 AM
Opalg
Quote:

Originally Posted by subfallen
Is it possible to find a function g(x) ∈ L1([0,1]) so that:

∫x^n g(x)dx = δ_{n1}

For n = 0, 1, 2,..., N where N is finite? (The right hand side denotes the Kronecker delta.)

It is even possible to find a polynomial of degree N with this property. The monomials $\displaystyle x^n\ (0\leqslant n\leqslant N)$ form a basis for the (N+1)-dimensional space of polynomials with degree at most N. The subspace spanned by $\displaystyle 1,\,x^2,\,x^3,\ldots,x^N$ has dimension N. Let f(x) be a nonzero element in its one-dimensional orthogonal complement with respect to the inner product $\displaystyle \langle u(x),v(x)\rangle = \int_0^1u(x)v(x)\,dx$ and let $\displaystyle g(x) = f(x)/\langle x,f(x)\rangle$.