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Quantum signal processing (QSP) represents a real scalar polynomial of degree $d$using a product of unitary matrices of size $2\times 2$, parameterized by $(d+1)$real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree $d\to \mathrm{\infty }$. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the ${\ell }^1$space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the ${\ell }^1$space. The algorithm uses only double precision arithmetic operations, and provably converges when the ${\ell }^1$norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of $d$. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit $d\to \mathrm{\infty }$.