All polynomials of order $p$ can be written as $\sum a_i (1 + t_i x)^p$? Given a bivariate function (called "kernel")
$$K(t,x) = (1 + tx)^p, t\in\mathbb{R}, x\in\mathbb{R}$$
I am looking at the space $H$ consists of
$$f(x) = \sum_{i=1}^n a_i (1 + t_i x)^p$$
It's obvious that the space $H$ is in the space of polynomials of order $p$, but how do I show the converse? Namely, is it true that for any polynomial of order $p$, there exists $a_1,\cdots, a_n$ and $t_1, \cdots, t_n$ such that
$$k_p x^p + \cdots + k_1 x + k_0 = \sum_{i=1}^n a_i \left(1 + t_i x\right)^p?$$
I tried to expand the right hand side and I don't think that helps.
 A: The Vandermonde determinant shows that for any $p+1$ distinct $t_i$, $(1+t_i x)^p$ are linearly independent, hence span the polynomials of degree $\leq p$. (This is equivalent to the comment by @Hellen.) To give a tiny bit of detail: For fixed $t_1,\dotsc,t_{p+1}$, the matrix whose entries are the coefficients of the $(1+t_i x)^p$, along rows say, is a slightly modified Vandermonde matrix: each entry is a $t_i^j$, times a binomial coefficient. The binomial factors are constant on each column. This affects the determinant of the matrix, but it doesn't change that the matrix still has full rank.
So, every polynomial $f$ can be written as a combination of $p+1$ of the $(1+tx)^p$'s (in fact, using any $p+1$ of them). By a very general result, for any $f$, there exist some $t_1,\dotsc,t_p$ ($n=p$ instead of $p+1$) s.t. the span of the $(1+t_i x)^p$ includes $f$: in a nutshell, a general hyperplane through $f$ cuts the curve $\{(1+tx)^p \mid t \in \mathbb{R}\}$ in $p$ points because that's the curve's degree; those points span the hyperplane, including $f$.
Certainly many $f$ require fewer than $p$ terms. (E.g., $f=(1+t_1 x)^p + (1+t_2 x)^p$ requires just $2$.) Examples like $f=x$ show (non-obviously) that $n=p$ can be required.
Further study of this, from an algebraic/geometric point of view is often referred to by terms like Waring rank and symmetric rank of symmetric tensors. If you're interested, some nice introductions include Reznick, On the length of binary forms, 2010 (over $\mathbb{C}$ and various subfields of $\mathbb{C}$), Reznick, Laws of inertia in higher degree binary forms, 2009 (over $\mathbb{R}$), and Carlini, et al, Four lectures on secant varieties, 2013. (There are certainly other points of view such as interpolation/sampling, but I don't know that terminology or literature, sorry.)
