Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

Effect of data encoding on the expressive power of variational quantum-machine-learning models

Abstract: Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

A First Course in Wavelets with Fourier Analysis

Abstract: 0. Inner Product Spaces. 1. Fourier Series. 2. The Fourier Transform. 3. Discrete Fourier Analysis. 4. Wavelet Analysis. 5. Multiresolution Analysis. 6. The Daubechies Wavelets. 7. Other Wavelet Topics. Appendix A. Technical Matters. Appendix B. Matlab Routines. Bibliography.

The cumulant theory of cyclostationary time-series. I. Foundation

Abstract: The problem of characterizing the sine-wave components in the output of a polynomial nonlinear system with a cyclostationary random time-series input is investigated. The concept of a pure nth-order sine wave is introduced, and it is shown that pure nth-order sine-wave strengths in the output time-series are given by scaled Fourier coefficients of the polyperiodic temporal cumulant of the input time-series. The higher order moments and cumulants of narrowband spectral components of time-series are defined and then idealized to the case of infinitesimal bandwidth. Such spectral moments and cumulants are shown to be characterized by the Fourier transforms of the temporal moments and cumulants of the time-series. It is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts. To put the contributions of the paper in perspective, a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided. >