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Implicit neural representations (INRs) have emerged as powerful tools for encoding signals, yet dominant MLP-based designs often suffer from slow convergence, and how their accuracy depends on their size is known only empirically. We introduce FUTON (Fourier Tensor Network), which models signals as generalized Fourier series whose coefficients are parameterized by a low-rank tensor decomposition, read out by a linear or MLP head. FUTON combines complementary inductive biases: Fourier bases capture smoothness and periodicity, while the low-rank parameterization enforces low-dimensional spectral structure. Beyond universal approximation, we show that with a linear head, the best FUTON's error splits exactly into the truncation of the signal's series and the best low-rank error of its coefficients, which we bracket from the samples before training, with a FUTON that attains the upper bound. On image fitting, occupancy volumes, and novel view synthesis, against twelve INRs of matched size, FUTON achieves the best average quality, reaches the final quality of each task's strongest baseline 4-34x sooner than that baseline, infers fastest, and trains in the least memory.