Let $B=B_\frac{1}{2}(0) \subset \mathbb{R}^n$. I want to show that given a function $F \in C_c(\mathbb{R}^n)$, with supp $F \subset B$ that the convolution $F \ast L_k$ restricted on $B$ is a polynomial of degree at most $k$.
My definition of the Landau kernel is $L_k: \mathbb{R}^n \longrightarrow \mathbb{R}$, $$ L_k(x)=\frac{1}{c^n_k} \prod_{i=1}^n (1-x_{i}^2)^k 1_{[-1,1]^n}(x), \quad \text{ where } \quad c_k = \int_{-1}^1 (1-t^2)^kdt, \quad k \in \mathbb{N}. $$
I have already gotten to
$$ (F \ast L_k)(x) = \int_{\text{supp }F} F(y)L_k(y-x)dy, $$
but seem to be stuck here. Any help would be greatly appreciated.
Hint: Just keep writing things out. Using your definition of $L_k$, we get that
\begin{align} F \ast L_k(x) & = \int_{supp F} F(y) \prod_{i=1}^n (1 - (x_i - y_i)^2)^k dy \end{align}
You can use binomial theorem to deal with $(1-(x_i - y_i)^2)^k $. The details are tedious but you get a linear combination of powers of $x_i$ with coffecients being integrals with respect to $y$.
E.g if the dimension $n=1$, we get that
\begin{align} F \ast L_k (x)& = \int_{supp F} F(y) (1-(x-y)^2)^k dy\\ & = \int_{suppF}F(y) \sum_{m=0}^k -1^{k} \binom{k}{m}(x-y)^{2m} dy\\ & \vdots \\ & \text{More details (i.e now expand the} (x-y)^{2m} \text{and take} \ x \ \text{out of the integral} ) \\ & \vdots \\ & = \sum_{i = 0}^{2k} x^i\int_{supp F} F(y) g_i(y) dy \end{align}
where $g_i(y)$ is some polynomial in the variable $y$. Therefore the integrals become coefficients for $x^i$ giving us that $F \ast L_k $ is a polynomial of degree $2k$ (this is true regardless of the support of $F \ast L_k $).