Fourier Extension Research Articles

A new approach to the Fourier extension problem for the paraboloid

A new approach to the Fourier extension problem for the paraboloid

Numerical differentiation of the piecewise smooth function by using Fourier extension method

Numerical differentiation of the piecewise smooth function is considered in this paper. To avoid the large error of numerical differentiation that may occur near potential non-smooth points, we identify the discontinuity points of the first or second derivative of the function. Then we divide the domain of the function into several sub-domains. For each sub-domain, the approximation is constructed by Fourier extension, and the global approximation of the piecewise smooth function is formed by superposition to improve accuracy. Some numerical experiments are conducted to further verify the efficacy of the method.

Tomographic Fourier extension identities for submanifolds of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}

We establish identities for the composition Tk,n(|gdσ^|2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{k,n}(|\\widehat{gd\\sigma }|^2)$$\\end{document}, where g↦gdσ^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\mapsto \\widehat{gd\\sigma }$$\\end{document} is the Fourier extension operator associated with a general smooth k-dimensional submanifold of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, and Tk,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{k,n}$$\\end{document} is the k-plane transform. Several connections to problems in Fourier restriction theory are presented.

A stationary set method for estimating oscillatory integrals

We propose a new method of estimating oscillatory integrals, which we call a “stationary set” method. We use it to obtain the sharp convergence exponents of Tarry’s problems in dimension two for every degree k\ge 2 . As a consequence, we obtain sharp L^{\infty} \to L^{p} Fourier extension estimates for a family of monomial surfaces.

A Generalized Iterated Tikhonov Method in the Fourier Domain for Determining the Unknown Source of the Time-Fractional Diffusion Equation

In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method.

Block-radial symmetry breaking for ground states of biharmonic NLS

We prove that the biharmonic NLS equation Δ2u+2Δu+(1+ε)u=|u|p-2uinRd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta ^2 u +2\\Delta u+(1+\\varepsilon )u=|u|^{p-2}u\\,\\,\\, in {\\mathbb {R}}^d \\end{aligned}$$\\end{document}has at least k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+1$$\\end{document} geometrically distinct solutions if ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} is small enough and 2<p<2⋆k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2<p<2_\\star ^k$$\\end{document}, where 2⋆k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2_\\star ^k$$\\end{document} is an explicit critical exponent arising from the Fourier restriction theory of O(d-k)×O(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(d-k)\ imes O(k)$$\\end{document}-symmetric functions. This extends the recent symmetry breaking result of Lenzmann–Weth (Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates, 2023) and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of k. We further prove that, as ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+$$\\end{document}, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.

Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates

We consider ground state solutions u ∈ H2(ℝN) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form Δ2u+2aΔu+bu−∣u∣p−2u=0inℝN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Delta ^2}u + 2a\\Delta u + bu - |u{|^{p - 2}}u = 0\\,\\,\\,\\,{\\rm{in}}\\,\\,{\\mathbb{R}^N}$$\\end{document} with positive constants a, b > 0 and exponents 2 < p < 2*, where 2∗=2NN−4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${2^ * } = {{2N} \\over {N - 4}}$$\\end{document} if N > 4 and 2* = ∞ if N ≤ 4. By exploiting a connection to the adjoint Stein–Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states u ∈ H2(ℝN) in dimension N ≥ 2 fail to be radially symmetric for all exponents 2<p<2N+2N−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 < p < {{2N + 2} \\over {N - 1}}$$\\end{document} in a suitable regime of a, b > 0.As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained L2-mass and for a related problem on the unit ball in ℝN subject to Dirichlet boundary conditions.

Oscillatory integral operators with homogeneous phase functions

AbstractOscillatory integral operators with 1-homogeneous phase functions satisfying a convexity condition are considered. For these we show the Lp–Lp-estimates for the Fourier extension operator of the cone due to Ou–Wang via polynomial partitioning. For this purpose, we combine the arguments of Ou–Wang with the analysis of Guth–Hickman–Iliopoulou, who previously showed sharp Lp–Lp-estimates for non-homogeneous phase functions with variable coefficients under a convexity assumption. Furthermore, we provide examples exhibiting Kakeya compression, which shows a more restrictive range than dictated by the Knapp example in higher dimensions. We apply the oscillatory integral estimates to show new local smoothing estimates for wave equations on compact Riemannian manifolds (M, g) with dim M ≥ 3. This generalizes the argument for the Euclidean wave equation due to Gao–Liu–Miao–Xi.

Sharp Fourier extension on the circle under arithmetic constraints

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a B3-set.

Environmental impact of foreign direct investment in Turkey: does the quality of institutions matter? Evidence from time series analysis using the Fourier extension.

Since the 1980s, Turkey has experienced a significant increase in both foreign direct investment (FDI) and its ecological footprint (EFP). While FDI is widely acknowledged as a pivotal driver of economic growth, its impact on environmental degradation is multifaceted and debated. Moreover, a country's institutional framework plays a key role in shaping this relationship. Yet, the influence of institutional structures on the FDI-environment nexus is often neglected in current literature. In this study, we investigate the environmental implications of FDI in Turkey from 1984 to 2018, employing time series analysis with a Fourier extension and accounting for institutional quality. Fourier function models give more effective results in modeling structural breaks. We first use Fourier techniques to assess the unit root and cointegration relationship. Upon establishing cointegration, we employ the DOLS estimator, extended with Fourier terms, to determine the long-term coefficients. We then assess the causal relationship between the variables using the Fourier causality test. Our findings indicate that while FDI exacerbates environmental degradation (supporting the pollution haven hypothesis), the interaction term of FDI-institutional quality mitigates this degradation (supporting the pollution halo hypothesis). Given these empirical findings, this study suggests that strengthening Turkey's institutional quality has the potential to amplify the environmental advantages of FDI, alongside its economic benefits.

Spectral projectors, resolvent, and Fourier restriction on the hyperbolic space

We develop a unified approach to proving Lp−Lq boundedness of spectral projectors, the resolvent of the Laplace-Beltrami operator and its derivative on Hd. In the case of spectral projectors, and when p and q are in duality, the dependence of the implicit constant on p is shown to be sharp. We also give partial results on the question of Lp−Lq boundedness of the Fourier extension operator. As an application, we prove smoothing estimates for the free Schrödinger equation on Hd and a limiting absorption principle for the electromagnetic Schrödinger equation with small potentials.

AAA interpolation of equispaced data

We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced samples. Although it is always better to approximate from large numbers of samples if they are available, whether equispaced or not, this method often performs impressively even when the sampling grid is coarse. In most cases it gives more accurate approximations than other methods. We support this claim with a review and discussion of nine classes of existing methods in the light of general properties of approximation theory as well as the “impossibility theorem” for equispaced approximation. We make careful use of numerical experiments, which are summarized in a sequence of nine figures. Among our new contributions is the observation, summarized in Fig. 7, that methods such as polynomial least-squares and Fourier extension may be either exponentially accurate and exponentially unstable, or less accurate and stable, depending on implementation.

Extremals for $$\alpha $$-Strichartz Inequalities

A necessary and sufficient condition on the precompactness of extremal sequences for one-dimensional $$\alpha $$ -Strichartz inequalities, equivalently $$\alpha $$ -Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $$\alpha $$ -Strichartz inequalities.

Efficient function approximation on general bounded domains using splines on a Cartesian grid

Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a regular but oversampled grid that is defined on a bounding box. This approach allows the use of high-order and highly structured splines as a basis for piecewise polynomials. The methodology is analogous to that of Fourier extensions, using Fourier series on a bounding box, which leads to spectral accuracy for smooth functions. However, Fourier extension approximations involve solving a highly ill-conditioned linear system, and this is an expensive step. The computational complexity of recent algorithms is $\mathcal {O}\left (N\log ^{2}\left (N\right )\right )$ in 1-D and $\mathcal {O}\left (N^{2}\log ^{2}\left (N\right )\right )$ in 2-D. We show that, compared to Fourier extension, the compact support of B-splines enables improved complexity for multivariate approximations, namely $\mathcal {O}(N)$ in 1-D, $\mathcal {O}\left (N^{3/2}\right )$ in 2-D and more generally $\mathcal {O}\left (N^{3(d-1)/d}\right )$ in d-D with d > 1. By using a direct sparse QR solver for a related linear system, we also observe that the computational complexity can be nearly linear in practice. This comes at the cost of achieving only algebraic rates of convergence. Our statements are corroborated with numerical experiments and Julia code is available.

Local Extension Estimates for the Hyperbolic Hyperboloid in Three Dimensions

We establish Fourier extension estimates for compact subsets of the hyperbolic hyperboloid in three dimensions via polynomial partitioning.

Band-Limited Maximizers for a Fourier Extension Inequality on the Circle, II

Among the class of functions on the circle with Fourier modes up to degree 120, constant functions are the unique real-valued maximizers for the endpoint Tomas-Stein inequality.

A fast rapidly convergent method for approximation of convolutions with applications to wave scattering and some other problems

In this article, we discuss an O(Nlog⁡N) rapidly convergent algorithm for the numerical approximation of the convolution integral with weakly singular kernels and compactly supported densities with possible jump discontinuities. To achieve the reduced computational complexity, we utilize the Fast Fourier Transform (FFT) on a uniform grid of size N for approximating the convolution. To facilitate this and maintain the accuracy, we primarily rely on a periodic Fourier extension of the density with a suitably large period depending on the support of the density. While the method's convergence rate improves with increasing smoothness of the periodic extension and, in fact, approximations exhibit super-algebraic convergence when the extension is infinitely differentiable, it converges only linearly when the density has jump discontinuities. In this context, we present two different procedures to enhance the convergence speed. Firstly, we utilize a certain Fourier smoothing technique to accelerate the convergence to achieve the quadratic rate in the overall approximation. Finally, to make the method truly high order, we augment the basic scheme by including a “thin” boundary grid and employing a specialized high-order boundary integrator. We validate its performance in terms of accuracy as well as computational efficiency through a variety of numerical experiments. In particular, to demonstrate the method's utility, we apply the integration scheme for the numerical solution of certain partial differential equations. Moreover, we also apply the quadrature to obtain a fast and high-order Nyström solver for the solution of the Lippmann-Schwinger integral equation.

Boundary condition limitation in an inverse source problem and its overcoming

In this paper, we point out the main disadvantage of the existing methods for solving an inverse source problems: The smoothness of the solutions is limited by the boundary condition. A Fourier extension method with a modified Tikhonov regularization is presented to overcome the limitation. Compared with previous studies, the convergence result of the new method is established under a weaker a priori assumption. Numerical experiments further verify the advantage of the method.

How Exponentially Ill-Conditioned Are Contiguous Submatrices of the Fourier Matrix?

Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT) matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such $p\times q$ submatrix of the $N\times N$ DFT matrix is at least $ \exp \left( \frac{\pi}{2} \left[\min(p,q)- \frac{pq}{N}\right]\right)$, up to algebraic prefactors. That is, fixing the shape parameters $(\al,\bt):=(p/N,q/N)\in(0,1)^2$, the growth is $e^{\rho N}$ as $N\to\infty$, the exponential rate being $\rho = \frac{\pi}{2}[\min(\alpha,\beta)- \alpha\beta]$. Our proof uses the Kaiser--Bessel transform pair (of which we give a self-contained proof), plus estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kernel $e^{ixt}$, we also prove another lower bound $(4/e\pi \al)^q$, up to algebraic prefactors, which is stronger than the above for small $\al$ and $\bt$. When combined, the bounds are within a factor of two of the empirical asymptotic rate, uniformly over $(0,1)^2$, and become sharp in certain regions. However, the results are not asymptotic: they apply to essentially all $N$, $p$, and $q$, and with all constants explicit.

On vanishing near corners of conductive transmission eigenfunctions

This paper is concerned with the geometric structure of the transmission eigenvalue problem associated with a general conductive transmission condition. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation is the Fourier extension of the transmission eigenfunction, and the growth rate of the density function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in Diao et al. (Commun Partial Differ Equ 46(4):630–679, 2021) and Blåsten and Liu (J Funct Anal 273:3616–3632, 2017) as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.