[math-fun] extend exp(x) for quickly convergent Fourier series

Define a function f(x) on the interval -1 <= x < 0, to create a function g(x), such that: 1. g(x)=f(x) for -1 <= x < 0 2. g(x)=exp(x) for 0 <= x <= 1 3. The period 2 Fourier series of g(x) on the interval -1 <= x <= 1, of the form: a[0] + sum(a[i]*cos(i*Pi*x),i=1..infinity) + sum(b[i]*sin(i*Pi*x),i=1..infinity) converges to exp(x) on the interval 0 <= x <= 1, faster than choosing f(x)=exp(x) Can you find an f(x) such that g(x) has the quickest converging Fourier series to exp(x) on [0,1] ?