Boundedness of an infinite series

Question

I have an expression $$f(x,y) = \sum^{\infty}_{n=0}\left({\rm A}_{n}x^{n}+{\rm B}_{n}x^{-n-1}\right){\rm P}_{n}(\cos{y})$$ where ${\rm P}_{n}(\cos{y})$ is the $n$'th order Legendre polynomial. I want to modify this so that it is bounded in $(x, y)\in[0,1]\times[0,\pi]$. How do I do this? I noticed that if $x=0$ then this isn't bounded so if we set ${\rm B}_{n} = 0$ then we have removed that problem. But I am not sure for $x=1$ and the values for $y$, can anyone give me some hints?

Answer 1

Setting $B_n = 0$ and that $\displaystyle \sum_{n=0}^{\infty} A_n$ is absolutely convergent should be sufficient. This is so because, setting $B_n=0$, we have $$f(x,y) = \sum_{n=0}^{\infty} A_nx^n P_n(\cos(y))$$ Since $\cos(y) \in [-1,1]$, we have $P_n(\cos(y)) \in [-1,1]$. Hence on $[0,1] \times [0,\pi]$, we have $$f(x,y) = \sum_{n=0}^{\infty} A_nx^n P_n(\cos(y)) \leq \sum_{n=0}^{\infty} \vert A_n \vert \vert x^n \vert \vert P_n(\cos(y)) \vert \leq \sum_{n=0}^{\infty} \vert A_n \vert$$ Hence, absolute convergence of $\displaystyle \sum_{n=0}^{\infty} A_n$ is sufficient.