Tag: Functional Analysis

The Fourier–Walsh expansion represents functions on $\Omega_n=\lbrace-1,1\rbrace^n$ in an orthonormal basis of coordinate products. The construction proceeds from a measure space to its Hilbert space $L^2$, then uses orthogonality to derive the coefficients and approximation formulas. On the Boolean cube, the integrals defining the inner product become finite averages.

$\quad$ We study representations of functions of inner products via embeddings into Hilbert spaces. While polynomials with nonnegative coefficients admit a single feature map realization, this approach fails when negative coefficients are present. To address this limitation, we introduce a two-map framework. Specifically, for any analytic function $f(x) = \sum_{k=0}^{\infty} a_k x^k$ with global convergence, we construct a Hilbert space $\mathcal{H}$ and transformations $\Phi, \Psi: \mathbb{R}^n \to \mathcal{H}$ such that $$\langle \Phi(u), \Psi(v) \rangle = f(\langle u, v \rangle) \quad \text{for all } u, v \in \mathbb{R}^n.$$