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On the stability of unevenly spaced samples for interpolation and quadrature

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Abstract

Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, the question of how the uneven distribution of the samples impacts the quality of interpolation and quadrature is analyzed. Starting with equally spaced nodes on \([-\pi ,\pi )\) with grid spacing h, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount \(\le \alpha h\), where \(0 \le \alpha < 1/2\) is a fixed constant. A discrete version of the Kadec-1/4 theorem is proved, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of h, for any \(\alpha <1/4\). Then, it is shown that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when \(0\le \alpha <1/4\). Though, quadrature rules at perturbed nodes can have negative weights for any \(\alpha >0\), a bound on the absolute sum of the quadrature weights is provided, which shows that perturbed equally spaced grids with small \(\alpha \) can be used without numerical woes. While the proof techniques work primarily when \(0 \le \alpha < 1/4\), it is shown that a small amount of oversampling extends our results to the case when \(1/4\le \alpha <1/2\).

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Notes

  1. For two functions \(g_1(N)\) and \(g_2(N)\), one writes \(g_1(N) = \varOmega (g_2(N))\) if there is a constant \(C>0\) that is independent of N such that \(g_1(N)\ge Cg_2(N)\) for all N.

  2. Note that \(\sum _{j =1}^{N}1/(j+\alpha )> \sum _{j=1}^N 1/(j+1) >\log (N) - 1/2\).

  3. If \(\alpha \ge (1-2\alpha _0)/2\), then one can pick some \(\alpha _1\) such that \(\alpha< \alpha _1 < \alpha _0\) and \(\alpha < (1-2\alpha _1)/2\). Since Eq. (5.2) holds if \(\alpha _0\) is replaced by \(\alpha _1\), it also holds for \(\alpha _0\).

  4. Note that we do not necessarily have \(\left| x_k - {\tilde{x}}_k\right| \le \alpha h\) for all k. Instead, we have \(\left| x_k - {\tilde{x}}_k\right| \le 2\alpha h\).

  5. Otherwise, swap the roles of \(k_1\) and \(k_2\).

  6. Note that j may not be between \(k_1\) and \(k_2\). However, the claim follows regardless of how j compares to \(k_1\) and \(k_2\).

References

  1. Adcock, B., Gataric, M., Hansen, A.: On stable reconstructions from nonuniform Fourier measurements. SIAM J. Imaging Sci. 7(3), 1690–1723 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Adcock, B., Hansen, A.C.: A generalized sampling theorem for stable reconstructions in arbitrary bases. J. Fourier Anal. Appl. 18(4), 685–716 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Anderson, B., Ash, J.M., Jones, R.L., Rider, D.G., Saffari, B.: Exponential sums with coefficients 0 or 1 and concentrated \(L^p\) norms. Ann. Inst. Fourier (Grenoble) 57(5), 1377–1404 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Austin, A.: Some New Results on and Applications of Interpolation in Numerical Computation. Mathematical Institute, University of Oxford, Oxford (2016)

    Google Scholar 

  5. Austin, A.P., Trefethen, L.N.: Trigonometric interpolation and quadrature in perturbed points. SIAM J. Numer. Anal. 55(5), 2113–2122 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Bagchi, S., Mitra, S.K.: The Nonuniform Discrete Fourier Transform and its Applications in Signal Processing, vol. 463. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  7. Barnett, A.H., Magland, J., afKlinteberg, L.: A parallel nonuniform fast Fourier transform library based on an “exponential of semicircle’’ kernel. SIAM J. Sci. Comput. 41(5), C479–C504 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Chui, C.K., Zhong, L.: Polynomial interpolation and Marcinkiewicz-Zygmund inequalities on the unit circle. J. Math. Anal. Appl. 233(1), 387–405 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297–301 (1965)

    MathSciNet  MATH  Google Scholar 

  10. Curtiss, J.H.: Polynomial interpolation in points equidistributed on the unit circle. Pacific J. Math. 12, 863–877 (1962)

    MathSciNet  MATH  Google Scholar 

  11. Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications, Oxford (2014)

    Google Scholar 

  12. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)

    MathSciNet  MATH  Google Scholar 

  13. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14(6), 1368–1393 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. II. Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Eckner, A.: Algorithms for unevenly-spaced time series: Moving averages and other rolling operators. In: Working Paper (2012)

  16. Euler, L.: Inventio summae cuiusque seriei ex dato termino generali. Commentarii academiae scientiarum Petropolitanae pp. 9–22 (1741)

  17. Feichtinger, H.G., Gröchenig, K.: Iterative reconstruction of multivariate band-limited functions from irregular sampling values. SIAM J. Math. Anal. 23(1), 244–261 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Feichtinger, H.G., Gröchenig, K., Strohmer, T.: Efficient numerical methods in non-uniform sampling theory. Numer. Math. 69(4), 423–440 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Filbir, F., Mhaskar, H.N.: A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel. J. Fourier Anal. Appl. 16(5), 629–657 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Gowrishankar, S., Basavarajuand, T., Manjaiah, D., Sarkar, S.: Issues in wireless sensor networks. Proceedings of the World Congress on Engineering (2008)

  21. Gröchenig, K.: A discrete theory of irregular sampling. Linear Algebra Appl. 193, 129–150 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Gröchenig, K.: Sampling, Marcinkiewicz-Zygmund inequalities, approximation, and quadrature rules. J. Approx. Theory 257, 105455 (2020). (20)

    MathSciNet  MATH  Google Scholar 

  23. Henrici, P.: Barycentric formulas for interpolating trigonometric polynomials and their conjugates. Numer. Math. 33(2), 225–234 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Hunter, D.B.: The evaluation of integrals of periodic analytic functions. Nordisk Tidskr. Informationsbehandling (BIT) 11, 175–180 (1971)

    MathSciNet  MATH  Google Scholar 

  25. Kadec, M.I.: The exact value of the Paley-Wiener constant. Dokl. Akad. Nauk SSSR 155, 1253–1254 (1964)

    MathSciNet  Google Scholar 

  26. Kämmerer, L., Ullrich, T., Volkmer, T.: Worst-case recovery guarantees for least squares approximation using random samples. Constr. Approx. 54(2), 295–352 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Kircheis, M., Potts, D.: Direct inversion of the nonequispaced fast Fourier transform. Linear Algebra Appl. 575, 106–140 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Kunis, S., Nagel, D., Strotmann, A.: Multivariate Vandermonde matrices with separated nodes on the unit circle are stable. Appl. Comput. Harmon. Anal. 58, 50–59 (2022)

    MathSciNet  MATH  Google Scholar 

  29. Kunis, S., Rolfes, J.: Another hilbert inequality and critically separated interpolation nodes. Proc. Appl. Math. Mech. 21(1), e202100,219 (2021)

    Google Scholar 

  30. Levinson, N.: Gap and Density Theorems. American Mathematical Society Colloquium Publications, vol. 26. American Mathematical Society, New York (1940)

    Google Scholar 

  31. Lim, B., Lee, J., Jang, J., Kim, K., Park, Y.J., Seo, K., Shim, Y.: Delayed output feedback control for gait assistance with a robotic hip exoskeleton. IEEE Trans. Rob. 35(4), 1055–1062 (2019)

    Google Scholar 

  32. Liu, L., Yin, S., Zhang, L., Yin, X., Yan, H.: Improved results on asymptotic stabilization for stochastic nonlinear time-delay systems with application to a chemical reactor system. IEEE Trans. Syst. Man Cybern. Syst. 47(1), 195–204 (2016)

    Google Scholar 

  33. Lubinsky, D.S.: Marcinkiewicz-Zygmund inequalities: methods and results. In: Recent Progress in Inequalities (Niš, 1996), Math. Appl., vol. 430, pp. 213–240. Kluwer Acad. Publ., Dordrecht (1998)

  34. Martinsson, P.G., Rokhlin, V., Tygert, M.: A fast algorithm for the inversion of general Toeplitz matrices. Comput. Math. Appl. 50(5–6), 741–752 (2005)

    MathSciNet  MATH  Google Scholar 

  35. Marzo, J., Pridhnani, B.: Sufficient conditions for sampling and interpolation on the sphere. Constr. Approx. 40(2), 241–257 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Marzo, J., Seip, K.: The Kadets 1/4 theorem for polynomials. Math. Scand. 104(2), 311–318 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comp. 70(235), 1113–1130 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math. Anal. 38(2), 574–594 (2006)

    MathSciNet  MATH  Google Scholar 

  39. Ortega-Cerdà, J., Saludes, J.: Marcinkiewicz-Zygmund inequalities. J. Approx. Theory 145(2), 237–252 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Paley, R.E.A.C., Wiener, N.: Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, vol. 19. American Mathematical Society, Providence, RI (1987). (Reprint of the 1934 original)

    Google Scholar 

  41. Piazzon, F., Vianello, M.: Stability inequalities for Lebesgue constants via Markov-like inequalities. Dolomites Res. Notes Approx. 11, 1–9 (2018)

    MathSciNet  Google Scholar 

  42. Pólya, G.: Über die Konvergenz von Quadraturverfahren. Math. Z. 37(1), 264–286 (1933)

    MathSciNet  MATH  Google Scholar 

  43. Potts, D., Steidl, G., Tasche, M.: Fast and stable algorithms for discrete spherical Fourier transforms. In: Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), vol. 275/276, pp. 433–450 (1998)

  44. Potts, D., Steidl, G., Tasche, M.: Fast Fourier transforms for nonequispaced data: a tutorial. In: Modern Sampling Theory, Applied and Numerical Harmonic Analysis, pp. 247–270. Birkhäuser Boston, Boston, MA (2001)

  45. Potts, D., Steidl, G., Tasche, M.: Numerical stability of fast trigonometric transforms-a worst case study. J. Concrete Appl. Math 1, 1–36 (2003)

    MathSciNet  MATH  Google Scholar 

  46. Potts, D., Tasche, M.: Uniform error estimates for nonequispaced fast Fourier transforms. Sampl. Theory Signal Process. Data Anal. 19(2), 42 (2021)

    MathSciNet  MATH  Google Scholar 

  47. Pujol-Vazquez, G., Mobayen, S., Acho, L.: Robust control design to the furuta system under time delay measurement feedback and exogenous-based perturbation. Mathematics 8(12) (2020)

  48. Quan, R.: Risk assessment of flood disaster in Shanghai based on spatial-temporal characteristics analysis from 251 to 2000. Environ. Earth Sci. 72(11), 4627–4638 (2014)

    Google Scholar 

  49. Rehfeld, K., Marwan, N., Heitzig, J., Kurths, J.: Comparison of correlation analysis techniques for irregularly sampled time series. Nonlinear Process. Geophys. 18(3), 389–404 (2011)

    Google Scholar 

  50. Ruiz-Antolín, D., Townsend, A.: A nonuniform fast Fourier transform based on low rank approximation. SIAM J. Sci. Comput. 40(1), A529–A547 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Runovski, K.V., Sickel, W.: Marcinkiewicz-Zygmund-type inequalities–trigonometric interpolation on non-uniform grids and unconditional Schauder bases in Besov spaces on the torus. Z. Anal. Anwendungen 16(3), 669–687 (1997)

    MathSciNet  MATH  Google Scholar 

  52. Scholes, M., Williams, J.: Estimating betas from nonsynchronous data. J. Financ. Econ. 5(3), 309–327 (1977)

    Google Scholar 

  53. Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37, 10–21 (1949)

    MathSciNet  Google Scholar 

  54. Stahl, F., Johansson, R.: Diabetes mellitus modeling and short-term prediction based on blood glucose measurements. Math. Biosci. 217(2), 101–117 (2009)

    MathSciNet  MATH  Google Scholar 

  55. Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014)

    MathSciNet  MATH  Google Scholar 

  56. Vio, R., Strohmer, T., Wamsteker, W.: On the reconstruction of irregularly sampled time series. Publ. Astron. Soc. Pac. 112(767), 74–90 (2000)

    Google Scholar 

  57. Wendel, J.G.: Note on the gamma function. Amer. Math. Monthly 55, 563–564 (1948)

    MathSciNet  Google Scholar 

  58. Wright, G.B., Javed, M., Montanelli, H., Trefethen, L.N.: Extension of Chebfun to periodic functions. SIAM J. Sci. Comput. 37(5), C554–C573 (2015)

    MathSciNet  MATH  Google Scholar 

  59. Yen, J.: On nonuniform sampling of bandwidth-limited signals. IRE Trans. Circuit Theory 3(4), 251–257 (1956)

    Google Scholar 

  60. Yen, J.: On the synthesis of line-sources and infinite strip-sources. IRE Trans. Antennas Propag. 5(1), 40–46 (1957)

    Google Scholar 

  61. Young, R.M.: An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, vol. 93. Academic Press Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1980)

    Google Scholar 

  62. Zappatore, A., Augieri, A., Bonifetto, R., Celentano, G., Savoldi, L., Vannozzi, A., Zanino, R.: Modeling quench propagation in the enea hts cable-in-conduit conductor. IEEE Trans. Appl. Supercond. 30(8), 1–7 (2020)

    Google Scholar 

  63. Zhang, R., Hredzak, B.: Distributed finite-time multiagent control for dc microgrids with time delays. IEEE Trans. Smart Grid 10(3), 2692–2701 (2019)

    Google Scholar 

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Acknowledgements

This work is partially supported by the National Science Foundation grants DMS-1818757, DMS-1952757 and DMS-2045646. We are thankful for many conversations with Anthony Austin, Nick Trefethen, and Kuan Xu regarding unevenly spaced trigonometric interpolation over several years. In private communication, Heather Wilber gave an initial proof of the discrete Kadec-1/4 theorem in December 2017, which we adapted for our purposes. It was Laurent Demanet who brought our attention to the conditioning of NUDFT matrices and we also benefited from Alex Barnett’s wisdom on the subject. The research direction became more focused after brain storming sessions during Cornell’s math REU program in 2021 and we thank Aparna Gupte, Yunan Yang, and Liu Zhang for discussions regarding the MZ inequalities and quadrature rules.

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This work is partially supported by the National Science Foundation Grants DMS-1818757, DMS-1952757 and DMS-2045646.

Appendices

Appendix A: A quadrature weight at perturbed nodes is negative

To show that the perturbed nodes in Eq. (4.1) have an associated quadrature weight that is negative when N is sufficiently large, we need a few trigonometric inequalities that are technical to derive.

Let \(p, q, r > 0\) be such that \(0< q-p-2r< q-p+2r < \pi \). The following trigonometric inequality holds:

$$\begin{aligned} \begin{aligned} \sin (p-r)\sin (q+r)&= \frac{\cos (q-p+2r)-\cos (p+q)}{2} \\&< \frac{\cos (q-p-2r)-\cos (p+q)}{2} = \sin (p+r)\sin (q-r). \end{aligned}\nonumber \\ \end{aligned}$$
(A.1)

The next lemma bounds the values of \(\ell _0\), the trigonometric Lagrange polynomial for \({\tilde{x}}_0\) associated with \(\{{\tilde{x}}_j\}_{j=-N}^N\) (see Eq. (4.3)), at an equally spaced node. This helps us bound the weight \({\tilde{w}}_0\) in Theorem 4.1.

Lemma A.1

Suppose N is even and \({\tilde{x}}_{-N},\ldots ,{\tilde{x}}_N\) are the perturbed nodes given in Eq. (4.1). We have

$$\begin{aligned} \left| \ell _0(kh)\right| = -\ell _0(kh) \ge \frac{\sin \left( \alpha h/2\right) }{\sin \left( (\left| k\right| + \alpha )h/2\right) }, \qquad -N \le k \le N,\quad k\ne 0, \end{aligned}$$
(A.2)

where \(h = 2\pi /(2N+1)\).

Proof

Since \(\ell _0(-x) = \ell _0(x)\) for all x, it suffices to show that Eq. (A.2) holds for \(k > 0\). By a simple counting argument, one can verify the equation \(\ell _0(kh) = \prod _{j=-N, j \ne 0}^N (\sin ((kh - {\tilde{x}}_j)/2) / \sin (-{\tilde{x}}_j/2)) \le 0\). Let \(s(x) = \sin (\left| xh/2\right| )\). Then, the numerator of \(\left| \ell _0(kh)\right| \) can be written as

$$\begin{aligned}&\prod _{\begin{array}{c} j = -N, \\ j \ne 0 \end{array}}^N \sin \left( \left| \frac{kh-{\tilde{x}}_{j}}{2}\right| \right) \\&\quad = \prod _{j=-N}^{-N+k-1} \sin \left( \left| \frac{kh-{\tilde{x}}_{j}}{2}\right| \right) \prod _{j=-N+k}^{-1} \sin \left( \left| \frac{kh-{\tilde{x}}_{j}}{2}\right| \right) \prod _{j=1}^{k} \sin \left( \left| \frac{kh-{\tilde{x}}_{j}}{2}\right| \right) \\&\qquad \prod _{j=k+1}^{N} \sin \left( \left| \frac{kh-{\tilde{x}}_{j}}{2}\right| \right) \\&\quad = \prod _{j=N-k+1}^N s(j + (-1)^{j+k}\alpha ) \prod _{j=k+1}^N s(j + (-1)^{j+k}\alpha ) \prod _{j=0}^{k-1} s(j + (-1)^{j+k+1}\alpha )\\&\qquad \prod _{j=1}^{N-k} s(j + (-1)^{j+k}\alpha ) \\&\quad = \prod _{j=1}^N s(j + (-1)^{j+k}\alpha ) \prod _{j=k+1}^N s(j + (-1)^{j+k}\alpha ) \prod _{j=0}^{k-1} s(j + (-1)^{j+k+1}\alpha ). \end{aligned}$$

Similarly, the denominator of \(\left| \ell _0(kh)\right| \) can be written as

$$\begin{aligned} \prod _{\begin{array}{c} j = -N, \\ j \ne 0 \end{array}}^N \sin \left( \left| \frac{{\tilde{x}}_j}{2}\right| \right) = \prod _{j=1}^N \sin ^2 \left( \left| \frac{{\tilde{x}}_j}{2}\right| \right) = \prod _{j=1}^N (s(j + (-1)^j\alpha ))^2. \end{aligned}$$

Using Eq. (A.1), we find the for every odd \(1 \le j \le N\), we have

$$\begin{aligned} s(j + \alpha )s((j+1) - \alpha ) \ge s(j - \alpha )s((j+1) + \alpha ). \end{aligned}$$
(A.3)

Hence, for every odd \(m_1\) and even \(m_2\) such that \(1\le m_1<m_2\le N\), Eq. (A.3) gives us

$$\begin{aligned} \prod _{j=m_1}^{m_2} s(j + (-1)^{j+1}\alpha ) \ge \prod _{j=m_1}^{m_2} s(j + (-1)^{j}\alpha ). \end{aligned}$$
(A.4)

We now consider the cases when k is even and odd separately.

Case I: \({\textbf{k}}\) is even. If k is even, then we have

$$\begin{aligned} -\ell _0(kh)&= \left| \ell _0(kh)\right| = \left( \prod _{j=1}^{k-2} \frac{s(j + (-1)^{j+1} \alpha )}{s(j + (-1)^{j} \alpha )}\right) \frac{s(k-1 + \alpha )}{s(k-1 - \alpha )} \frac{s(\alpha )}{s(k+\alpha )}\\&\ge \frac{s(\alpha )}{s(k+\alpha )} = \frac{\sin (\alpha h/2)}{\sin ((k+\alpha )h/2)}, \end{aligned}$$

where we used Eq. (A.4) with \(m_1 = 1\) and \(m_2 = k-2\).

Case II: \({\textbf{k}}\) is odd. If k is odd, then we have the following inequality on \(-\ell _0(kh)\):

$$\begin{aligned} -\ell _0(kh)&= \left| \ell _0(kh)\right| \ge \left( \prod _{j=k+2}^{N} \frac{s(j + (-1)^{j+1} \alpha )}{s(j + (-1)^{j} \alpha )}\right) \frac{s(k+1 - \alpha )}{s(k+1 + \alpha )} \frac{s(\alpha )}{s(k-\alpha )} \\&\ge \frac{s(k+1 - \alpha )}{s(k+1 + \alpha )} \frac{s(\alpha )}{s(k-\alpha )} \ge \frac{s(\alpha )}{s(k+\alpha )} = \frac{\sin (\alpha h/2)}{\sin ((k+\alpha )h/2)}, \end{aligned}$$

where the sequence of three inequalities are obtained from Eq. (A.4) by setting (1) \(m_1 = 1\) and \(m_2 = N\), (2) \(m_1 = k+2\) and \(m_2 = N\), and (3) \(m_1 = k\) and \(m_2 = k+1\), respectively. The proof is complete. \(\square \)

Appendix B: Only quadrature rules at ample perturbed nodes can have negative weights

For any \(0<\alpha <1/2\), we define \(N^{{\textrm{neg}}}_\alpha \) to be the smallest integer such that for \(N = N^{{\textrm{neg}}}_\alpha \) there exists a set of \(\alpha \)-perturbed quadrature nodes \(\{{\tilde{x}}_j\}_{j=-N}^{N}\) so that the associated exact quadrature rule on \({\mathscr {T}}_N\) has a negative weight. By Theorem 4.1, we know that \(N^{{\textrm{neg}}}_\alpha \) is finite for every \(\alpha \). Here, we derive an implicit lower bound Eq. (B.3) on \(N^{{\textrm{neg}}}_\alpha \) and provide a closed formula in Eq. (B.5), which proves \(\log ( N^{{\textrm{neg}}}_\alpha ) = \varTheta (\alpha ^{-1})\) as \(\alpha \rightarrow 0\). This is a rather technical result.

First, we define an equivalence relation on the indices. Let \(-N \le j, k \le N\) and define \(d(j,k) = \min \{\left| j-k\right| ,2N+1-\left| j-k\right| \}\). For a fixed j, define an equivalence class on \(\{-N, \ldots , j-1, j+1, \ldots , N\}\) by \(k_1 \sim _j k_2\) if and only if \(d(j,k_1) = d(j,k_2)\) and denote the equivalence class by \(S^j_{d(j,k_1)}\). Note that each of \(S^j_1, \ldots , S^j_N\) contains exactly 2 elements.

Lemma B.1

Let \(0\le \alpha < 1/2\), \(h = 2\pi /(2N+1)\), and \(\{{\tilde{x}}_k\}_{k=-N}^{N}\) be a set of \(\alpha \)-perturbed nodes. For fixed \(-N\le j\le N\) and for any \(-N \le k \le N\) with \(k \ne j\), we have

$$\begin{aligned} \left| \frac{\prod _{m=-N,m \ne j,k}^{N} \sin \left( \frac{x_{k} - {\tilde{x}}_m}{2}\right) }{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) }\right| \le \frac{1}{\sin \left( \frac{d(j,k)h}{2}\right) } \prod _{m=1}^N \frac{\sin ^2\left( \frac{mh + \alpha h}{2}\right) }{\sin \left( \frac{mh - 2\alpha h}{2}\right) \sin \left( \frac{mh}{2}\right) }, \end{aligned}$$
(B.1)

where \(x_k = {\tilde{x}}_j + (k-j)h\).Footnote 4

Proof

Let \(d = d(j,k)\) and \(j'\) be the element in \(S^k_{d}\) that is not equal to j. Then, we can write the numerator and denominator of Eq. (B.1) as

$$\begin{aligned} A&= \prod _{\begin{array}{c} m=-N,\\ m \ne j,k \end{array}}^{N} \sin \left( \frac{x_{k} - {\tilde{x}}_m}{2}\right) = \sin \left( \frac{x_k - {\tilde{x}}_{j'}}{2}\right) \prod _{\begin{array}{c} m=1, m \ne d, \\ S^k_m = \{k_1, k_2\} \end{array}}^{N} \left[ \sin \left( \frac{x_{k} - {\tilde{x}}_{k_1}}{2}\right) \sin \left( \frac{x_{k} - {\tilde{x}}_{k_2}}{2}\right) \right] ,\\ B&= \prod _{\begin{array}{c} m=-N,\\ m\ne j \end{array}}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) = \prod _{\begin{array}{c} m=1,\\ S^j_m = \{k_1, k_2\} \end{array}}^{N} \left[ \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_{k_1}}{2}\right) \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_{k_2}}{2}\right) \right] . \end{aligned}$$

Suppose \(S_m^k = \{k_1,k_2\}\), where \(1 \le m \le N\). Then, the distance between the two nodes satisfies \(d_{\mathbb {T}}(e^{ix_k}, e^{i{\tilde{x}}_{k_1}}) + d_{\mathbb {T}}(e^{ix_k}, e^{i{\tilde{x}}_{k_2}}) \le 2mh + 2\alpha h\) and \(d_{\mathbb {T}}(e^{ix_k}, e^{i{\tilde{x}}_{k_1}}), d_{\mathbb {T}}(e^{ix_k}, e^{i{\tilde{x}}_{k_2}}) \le mh+2\alpha h\), where \(d_{\mathbb {T}}(e^{ix}, e^{iy}) = \min \{\left| x-y\right| , 2\pi - \left| x-y\right| \}\) is the “distance" between \(e^{ix}\) and \(e^{iy}\) on the unit circle for \(x,y\in [-\pi ,\pi )\). Therefore, by elementary calculus, we have

$$\begin{aligned} \left| \sin \left( \frac{x_{k} - {\tilde{x}}_{k_1}}{2}\right) \sin \left( \frac{x_{k} - {\tilde{x}}_{k_2}}{2}\right) \right| \le \sin ^2 \left( \frac{mh+\alpha h}{2}\right) , \end{aligned}$$

where the lefthand side of the inequality above is maximized when \(d_{\mathbb {T}}(e^{ix_k}, e^{i{\tilde{x}}_{k_1}}) = d_{\mathbb {T}}(e^{ix_k}, e^{i{\tilde{x}}_{k_2}}) = mh+\alpha h\). Similarly, suppose \(S_m^j = \{k_1,k_2\}\). We have \(d_{\mathbb {T}}(e^{i{\tilde{x}}_j}, e^{i{\tilde{x}}_{k_1}}) + d_{\mathbb {T}}(e^{i{\tilde{x}}_j}, e^{i{\tilde{x}}_{k_2}}) \ge 2mh-2\alpha h\) and \(d_{\mathbb {T}}(e^{i{\tilde{x}}_j}, e^{i{\tilde{x}}_{k_1}}), d_{\mathbb {T}}(e^{i{\tilde{x}}_j}, e^{i{\tilde{x}}_{k_2}}) \ge mh-2\alpha h\). Hence,

$$\begin{aligned} \left| \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_{k_1}}{2}\right) \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_{k_2}}{2}\right) \right| \ge \sin \left( \frac{mh - 2\alpha h}{2}\right) \sin \left( \frac{mh}{2}\right) , \end{aligned}$$

where the lefthand side of the inequality above is minimized when \(d_{\mathbb {T}}(e^{i{\tilde{x}}_j}, e^{i{\tilde{x}}_{k_i}}) = mh\) and \(d_{\mathbb {T}}(e^{i{\tilde{x}}_j}, e^{i{\tilde{x}}_{k_{3-i}}}) = mh - 2\alpha h\) for \(i = 1\) or 2. This gives us

$$\begin{aligned} \left| \frac{A}{B}\right| \le \frac{\sin \left( \frac{dh + 2\alpha h}{2}\right) \prod _{m=1, m \ne d}^{N} \sin ^2\left( \frac{mh+\alpha h}{2}\right) }{\prod _{m=1}^N \sin \left( \frac{mh - 2\alpha h}{2}\right) \sin \left( \frac{mh}{2}\right) } \le \frac{1}{\sin \left( \frac{dh}{2}\right) } \prod _{m=1}^N \frac{\sin ^2\left( \frac{mh+\alpha h}{2}\right) }{\sin \left( \frac{mh - 2\alpha h}{2}\right) \sin \left( \frac{mh}{2}\right) }, \end{aligned}$$

as desired. \(\square \)

It is worth observing that A/B in the proof of Lemma B.1 is almost the Lagrange polynomial at \(x_j\). This connection is made precise in the next lemma.

Lemma B.2

Using the same notation as Lemma B.1, if \(\ell _j\) is the jth trigonometric Lagrange basis polynomial for \({\tilde{x}}_j\) associated with \(\{{\tilde{x}}_k\}_{k=-N}^{N}\), then

$$\begin{aligned} \left| \ell _j(x_{k_1}) + \ell _j(x_{k_2})\right| \le \frac{\pi \alpha }{d} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m}, \end{aligned}$$
(B.2)

where \(S^j_d = \{k_1, k_2\}\).

Proof

Let \(d = d(j,k)\) and suppose that \(1 \le d \le N\). For \(i = 1, 2\), we have

$$\begin{aligned} \ell _j(x_{k_i})&= \frac{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{x_{k_i} - {\tilde{x}}_m}{2}\right) }{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) } = \frac{\sin \left( \frac{x_{k_i} - {\tilde{x}}_{k_i}}{2}\right) \prod _{m=-N,m\ne j, k_i}^{N} \sin \left( \frac{x_{k_i} - {\tilde{x}}_m}{2}\right) }{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) }. \end{aligned}$$

By Lemma B.1, we find that

$$\begin{aligned} \left| \frac{\prod _{m=-N,m\ne j, k_i}^{N} \sin \left( \frac{x_{k_i} - {\tilde{x}}_m}{2}\right) }{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) }\right| \le \frac{\pi }{dh} \prod _{m=1}^N \frac{(mh + \alpha h)^2}{(mh - 2\alpha h)(mh)} = \frac{\pi }{dh} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m}, \end{aligned}$$

where we need the fact that \((2/\pi )x \le \sin x\) for all \(0 \le x \le \pi /2\) and the fact that \(\sin y / \sin x \le y / x\) for all \(0< x \le y < \pi \). We now prove Eq. (B.2) by considering the cases \(\ell _j(x_{k_1})\ell _j(x_{k_2}) \le 0\) and \(\ell _j(x_{k_1})\ell _j(x_{k_2}) > 0\) separately.

Case 1: \({{\varvec{\ell }}}_{\textbf{j}}(\textbf{x}_{\textbf{k}_{\textbf{1}}}){{\varvec{\ell }}}_{\textbf{j}}(\textbf{x}_{\textbf{k}_{\textbf{2}}}) \le \textbf{0}\). Since \(d_{\mathbb {T}}(e^{ix_{k_i}}, e^{i{\tilde{x}}_{k_i}}) \le 2 \alpha h\) for \(i = 1, 2\), we have

$$\begin{aligned}&\left| \ell _j(x_{k_1}) + \ell _j(x_{k_2})\right| \le \max \{\left| \ell _j(x_{k_1})\right| , \left| \ell _j(x_{k_2})\right| \} \\&\le \max _{i = 1, 2}\left[ \frac{d_{\mathbb {T}}(e^{ix_{k_i}}, e^{i{\tilde{x}}_{k_i}})}{2}\right] \frac{\pi }{dh} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m} \le \frac{\pi \alpha }{d} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m}, \end{aligned}$$

where we used the fact that \(\sin (x) \le x\) for all \(0 \le x \le \pi /2\).

Case 2: \({\varvec{\ell }}_{\textbf{j}}(\textbf{x}_{\textbf{k}_{\textbf{1}}}){{\varvec{\ell }}}_{\textbf{j}}(\textbf{x}_{\textbf{k}_{\textbf{2}}}) > \textbf{0}\). We claim that

$$\begin{aligned} \frac{\prod _{m=-N,m\ne j, k_1}^{N} \sin \left( \frac{x_{k_1} - {\tilde{x}}_m}{2}\right) }{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) } \frac{\prod _{m=-N,m\ne j, k_2}^{N} \sin \left( \frac{x_{k_2} - {\tilde{x}}_m}{2}\right) }{\prod _{m=-N,m\ne j}^{N} \sin \left( \frac{{\tilde{x}}_j - {\tilde{x}}_m}{2}\right) } < 0. \end{aligned}$$

The claim follows because if we assume, without loss of generality,Footnote 5 that \(k_1 < k_2\), then there are an even number of integers m, not including j, such that \(k_1< m < k_2\).Footnote 6 When \(m \ne k_1, k_2\), the signs of \(\sin ((x_{k_1}-{\tilde{x}}_m)/2)\) and \(\sin ((x_{k_2}-{\tilde{x}}_m)/2)\) are different if and only if \(k_1< m < k_2\). The claim follows from the fact that \(\sin ((x_{k_2}-{\tilde{x}}_{k_1})/2) \sin ((x_{k_1}-{\tilde{x}}_{k_2})/2) < 0\) because the remaining terms multiply to a positive number. Hence, we have \(\sin ((x_{k_1} - {\tilde{x}}_{k_1})/2) \sin ((x_{k_2} - {\tilde{x}}_{k_2})/2) < 0\) so that \((x_{k_1} - {\tilde{x}}_{k_1})(x_{k_2} - {\tilde{x}}_{k_2}) < 0\). Let \(\delta := {\tilde{x}}_j - jh\). Then, \(x_{k_i} - {\tilde{x}}_{k_i} < \alpha h + \delta \) if \(x_{k_i} - {\tilde{x}}_{k_i} > 0\) and \({\tilde{x}}_{k_i} - x_{k_i} < \alpha h - \delta \) if \(x_{k_i} - {\tilde{x}}_{k_i} < 0\). Hence, \((x_{k_1} - {\tilde{x}}_{k_1})(x_{k_2} - {\tilde{x}}_{k_2}) < 0\) implies \(d_{\mathbb {T}}(e^{ix_{k_1}}, e^{i{\tilde{x}}_{k_1}}) + d_{\mathbb {T}}(e^{ix_{k_2}}, e^{i{\tilde{x}}_{k_2}}) \le 2\alpha h\). By elementary calculus, we find that

$$\begin{aligned} \left| \sin \left( \frac{x_{k_1} - {\tilde{x}}_{k_1}}{2}\right) \right| + \left| \sin \left( \frac{x_{k_2} - {\tilde{x}}_{k_2}}{2}\right) \right| \le 2\sin \left( \frac{\alpha h}{2}\right) \le \alpha h. \end{aligned}$$

Putting this together, we obtain

$$\begin{aligned} \left| \ell _j(x_{k_1}) + \ell _j(x_{k_2})\right|&= \left| \ell _j(x_{k_1})\right| + \left| \ell _j(x_{k_2})\right| \\&\le \left( \left| \sin \left( \frac{x_{k_1} - {\tilde{x}}_{k_1}}{2}\right) \right| + \left| \sin \left( \frac{x_{k_2} - {\tilde{x}}_{k_2}}{2}\right) \right| \right) \frac{\pi }{dh} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m} \\&\le \frac{\pi \alpha }{d} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m}, \end{aligned}$$

as desired. \(\square \)

We are now ready to prove a lower bound on \(N^{{\textrm{neg}}}_\alpha \).

Theorem B.1

We have

$$\begin{aligned} g(N^{{\textrm{neg}}}_\alpha ) > \frac{1}{\pi \alpha }, \qquad g(N):= \prod _{m=1}^{N} \left[ \frac{(m + \alpha )^2}{(m - 2\alpha )m}\right] \left[ \sum _{d=1}^{N} \frac{1}{d}\right] \end{aligned}$$
(B.3)

for \(0<\alpha <1/2\).

Proof

We define \(g: {\mathbb {N}}\rightarrow {\mathbb {R}}\) as in Eq. (B.3). It suffices to show that \(g(N) \le 1/(\pi \alpha )\) implies that a quadrature rule at \(\alpha \)-perturbed nodes of degree N contains no negative weight. Let \(\{{\tilde{x}}_j\}_{j=-N}^{N}\) be a set of \(\alpha \)-perturbed nodes. Let \({\tilde{w}}_j\) be the quadrature weight associated with \({\tilde{x}}_j\) and let \(\ell _j\) be the corresponding trigonometric Lagrange basis polynomial. Let \(\{x_k\}_{k=-N}^{N}\), which may depend on j, be defined as in Lemma B.1. Then, we have \({\tilde{w}}_j = (2\pi /(2N+1))\sum _{k=-N}^{N} \ell _j(x_k)\). By Lemma B.2, we have

$$\begin{aligned}&\sum _{k=-N}^{N} \ell _j(x_k) = \ell _j(x_j) + \sum _{k=-N, k \ne j}^{N} \ell _j(x_k) \ge 1 - \sum _{d=1, S^j_d=\{k_1,k_2\}}^{N} \left| \ell _j(x_{k_1}) + \ell _j(x_{k_2})\right| \\&\ge 1 - \sum _{d=1}^{N} \left[ \frac{\pi \alpha }{d} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m}\right] = 1-\pi \alpha \prod _{m=1}^N \left[ \frac{(m + \alpha )^2}{(m - 2\alpha )m}\right] \left[ \sum _{d=1}^N \frac{1}{d}\right] \ge 0. \end{aligned}$$

This proves all quadrature weights are non-negative. \(\square \)

For small \(\alpha \), we can make the statement in Theorem B.1 more explicit.

Corollary B.1

For \(0<\alpha <0.15\) and a real number \(L > 0\) such that

$$\begin{aligned} (\alpha + L)e^{4L} \le \frac{\varGamma (1+\alpha )^2}{\pi \varGamma (1-2\alpha )}, \end{aligned}$$
(B.4)

where \(\varGamma \) is the gamma function, we find that \(N^{{\textrm{neg}}}_\alpha \) satisfies the following inequality:

$$\begin{aligned} \log (N^{{\textrm{neg}}}_\alpha +1+\alpha ) + \frac{1}{2N^{{\textrm{neg}}}_\alpha -1} > (1-\gamma ) + \frac{L}{\alpha }, \end{aligned}$$
(B.5)

where \(\gamma \approx 0.57722\) is the Euler–Mascheroni constant [16].

Proof

We aim to show that Eq. (B.4) implies Eq. (B.5). First, we use Gautschi’s inequality [57] to obtain

$$\begin{aligned}{} & {} \prod _{m=1}^N \frac{(m + \alpha )^2}{(m - 2\alpha )m} = \frac{\varGamma (1 - 2\alpha ) \varGamma (N + 1 + \alpha )^2}{\varGamma (N + 1 - 2\alpha )\varGamma (N + 1)\varGamma (1 + \alpha )^2}\nonumber \\{} & {} \le \frac{\varGamma (1 - 2\alpha )}{\varGamma (1 + \alpha )^2} (N + 1 + \alpha )^{4\alpha }. \end{aligned}$$
(B.6)

Assume that \(\log (N+1+\alpha ) + 1/(2N-1) \le (1-\gamma ) + L/\alpha \), for an integer \(N>0\). We have \(\log (N+1+\alpha ) \le L/\alpha \). Hence, \(N+1+\alpha \le e^{L/\alpha }\). By Eq. (B.6), we find that

$$\begin{aligned} g(N)\le & {} \frac{\varGamma (1 - 2\alpha )}{\varGamma (1 + \alpha )^2} (N + 1 + \alpha )^{4\alpha } \left( \log (N) + \gamma + \frac{1}{2N - 1}\right) \\{} & {} \le \frac{\varGamma (1 - 2\alpha )}{\varGamma (1 + \alpha )^2} e^{4L} \left( 1 + \frac{L}{\alpha }\right) \\\le & {} \frac{1}{\pi \alpha }, \end{aligned}$$

where we used the fact that \(\sum _{d=1}^N d^{-1} \le \log (N) + \gamma + 1/(2N-1) \le 1+L/\alpha \) [16] and the last inequality follows from Eq. (B.4). Since g(N) is an increasing function of N, we know from Eq. (B.3) that \(N < N^{{\textrm{neg}}}_\alpha \). Moreover, this holds for all \(N > 1\) that satisfies \(\log (N+1+\alpha ) + 1/(2N-1) \le (1-\gamma ) + L/\alpha \). When \(\alpha < 0.15\), we see that \(N^{{\textrm{neg}}}_\alpha \ge 2\) by Eq. (B.3) and this proves Eq. (B.5). \(\square \)

Corollary B.1 implies that \(\log (N^{{\textrm{neg}}}_\alpha ) = \varOmega (\alpha ^{-1})\) and by Theorem 4.1, we conclude that \(\log ( N^{{\textrm{neg}}}_\alpha ) = \varTheta (\alpha ^{-1})\) as \(\alpha \rightarrow 0\).

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Yu, A., Townsend, A. On the stability of unevenly spaced samples for interpolation and quadrature. Bit Numer Math 63, 23 (2023). https://doi.org/10.1007/s10543-023-00965-z

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