Pointwise and uniform convergence of Fourier extensions

Webb, Marcus; Coppé, Vincent; Huybrechs, Daan

doi:10.48550/arxiv.1811.09527

Cited by 2 publications

(2 citation statements)

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“…It is known [28] that the Fourier series of an analytical, periodic function converges exponentially fast, i.e, the error ε made in the approximation goes asymptotically as ε = − , c > 0, where q is the truncation order. Our goal is to obtain a periodic extension of f which is analytic in the whole interval [−π, π].…”

Section: Analytic-extension Fourier Seriesmentioning

confidence: 99%

Fourier-based quantum signal processing

Silva¹,

Borges²,

Aolita³

2022

Preprint

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Implementing general functions of operators is a powerful tool in quantum computation. It can be used as the basis for a variety of quantum algorithms including matrix inversion, real and imaginary-time evolution, and matrix powers. Quantum signal processing is the state of the art for this aim, assuming that the operator to be transformed is given as a block of a unitary matrix acting on an enlarged Hilbert space. Here we present an algorithm for Hermitian-operator function design from an oracle given by the unitary evolution with respect to that operator at a fixed time. Our algorithm implements a Fourier approximation of the target function based on the iteration of a basic sequence of single-qubit gates, for which we prove the expressibility. In addition, we present an efficient classical algorithm for calculating its parameters from the Fourier series coefficients. Our algorithm uses only one qubit ancilla regardless the degree of the approximating series. This contrasts with previous proposals, which required an ancillary register of size growing with the expansion degree. Our methods are compatible with Trotterised Hamiltonian simulations schemes and hybrid digital-analog approaches.

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Section: Analytic-extension Fourier Seriesmentioning

confidence: 99%

Fourier-based quantum signal processing

Silva¹,

Borges²,

Aolita³

2022

Preprint

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“…Obtaining L ∞ bounds from a discrete L 2 construction is a delicate issue which we address in this paper. When the discrete inner product (1.6) is replaced by the usual L 2 inner product, the analysis is simplified greatly and an exponential bound on the L ∞ norm of the difference f − q was obtained in [1,Theorem 2.3], see also [18,26]. Recent works have distinguished between the discrete Fourier extension defined in terms of the discrete inner product (1.6) and the continuous Fourier extension defined in terms of the usual L 2 inner product [1,2].…”

Section: Introductionmentioning

confidence: 99%

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

Geronimo

Liechty

2020

Advances in Mathematics

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The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann-Hilbert methods.

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Pointwise and uniform convergence of Fourier extensions

Cited by 2 publications

References 0 publications

Fourier-based quantum signal processing

Fourier-based quantum signal processing

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

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