Multivariable Fourier optical circuit synthesis

Determine whether and how the two-mode linear-optical Fourier architecture can be extended to synthesize multivariable functions of the form $f(x,y)=\sum_{s,t}w_{st}e^{i(sx+ty)}$, given that single-variable Fejér–Riesz and spectral-factorization methods do not guarantee a solution in the multivariable case.

Background

The paper establishes an exact construction for arbitrary finite univariate Fourier series using a two-mode passive linear-optical circuit, with an auxiliary polynomial obtained through Fejér–Riesz spectral factorization. The authors identify multivariable extension as a natural unresolved direction because the factorization tools that guarantee completion in one variable do not generally provide an analogous solution for multivariable trigonometric polynomials.

The unresolved issue concerns both whether a suitable optical circuit exists for multivariable Fourier representations and how such a circuit could be synthesized. Resolving it would extend the architecture from one-dimensional nonlinear functions to functions with multiple input variables.

References

The natural extension to this work is whether, and how, one may do the same for multivariable functions $f(x,y)$. This would require a circuit synthesis over a Fourier series of the form $f(x,y) = \sum_{s,t}w_{st}e{i(sx + ty)}$, for which the Fejér-Riesz and spectral factorisation methods no longer guarantee a solution as in the single variable case.

— Optical Fourier Architecture for Universal Nonlinear Functions  (2609.37746 - Mauser et al., 29 Sep 2026) in Section Discussion