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August  2020, 3(3): 157-163. doi: 10.3934/mfc.2020015

Summation of Gaussian shifts as Jacobi's third Theta function

Shengxin Zhu ,

Laboratory for Intelligent Computing and Financial Technology, Department of Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou, 215123, China

* Corresponding author: Shengxin Zhu

Received  December 2019 Revised  May 2020 Published  June 2020

Fund Project: This research is supported by Foundation of LCP(6142A05180501), Jiangsu Science and Technology Basic Research Program (BK20171237), Key Program Special Fund of XJTLU (KSF-E-21, KSF-P-02), Research Development Fund of XJTLU (RDF-2017-02-23), and partially supported by NSFC (No.11771002, 11571047, 11671049, 11671051, 6162003, and 11871339)

A proper choice of parameters of the Jacobi modular identity (Jacobi Imaginary transformation) implies that the summation of Gaussian shifts on infinity periodic grids can be represented as the Jacobi's third Theta function. As such, connection between summation of Gaussian shifts and the solution to a Schrödinger equation is explicitly shown. A concise and controllable upper bound of the saturation error for approximating constant functions with summation of Gaussian shifts can be immediately obtained in terms of the underlying shape parameter of the Gaussian. This sheds light on how to choose a shape parameter and provides further understanding on using Gaussians with increasingly flatness.

Keywords: Gaussian radial basis functions, Jacobi Theta function, modular identity, Jacobi's imaginary transformation, saturation error.
Mathematics Subject Classification: Primary: 65D05; Secondary: 65D05.
Citation: Shengxin Zhu. Summation of Gaussian shifts as Jacobi's third Theta function. Mathematical Foundations of Computing, 2020, 3 (3) : 157-163. doi: 10.3934/mfc.2020015
References:
[1]

B. J. C. Baxter, Norm estimates for inverses of Toeplitz distance matrices, J. Approx. Theory, 79 (1994), 222-242.  doi: 10.1006/jath.1994.1126.  Google Scholar

[2]

B. J. C. Baxter and N. Sivakumar, On shifted cardinal interpolation by Gaussians and multiquadrics, J. Approx. Theory, 87 (1996), 36-59.  doi: 10.1006/jath.1996.0091.  Google Scholar

[3]

R. Bellman, A Brief Introduction to Theta Functions, Athena Series: Selected Topics in Mathematics Holt, Rinehart and Winston, New York, 1961.  Google Scholar

[4]

J. P. Boyd, Error saturation in Gaussian radial basis functions on a finite interval, J. Comput. Appl. Math., 234 (2010), 1435-1441.  doi: 10.1016/j.cam.2010.02.019.  Google Scholar

[5]

J. P. Boyd and L. Wang, An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice, Appl. Math. Comput., 215 (2009), 2215-2223.  doi: 10.1016/j.amc.2009.08.037.  Google Scholar

[6]

M. D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Constr. Approx., 6 (1990), 225-255.  doi: 10.1007/BF01890410.  Google Scholar

[7] M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics, 12. Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511543241.  Google Scholar
[8]

A. Chernih and S. Hubbert, Closed form representations and properties of the generalised Wendland functions, J. Approx. Theory, 177 (2014), 17-33.  doi: 10.1016/j.jat.2013.09.005.  Google Scholar

[9]

Y. Choie and Y. Taguchi, A simple proof of the modular identity for theta series, Proc. Amer. Math. Soc., 133 (2005), 1935-1939.  doi: 10.1090/S0002-9939-05-07723-3.  Google Scholar

[10]

W. Couwenberg, A simple proof of the modular identity for theta functions, Proc. Amer. Math. Soc., 131 (2003), 3305-3307.  doi: 10.1090/S0002-9939-03-06902-8.  Google Scholar

[11]

T. A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions, Radial basis functions and partial differential equations, Comput. Math. Appl., 43 (2002), 413-422.  doi: 10.1016/S0898-1221(01)00295-4.  Google Scholar

[12]

G. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6437.  Google Scholar

[13]

G. Fasshauer, F. Hickernel and H. Woniakowski, On Dimension-independent Rates of Convergence for Function Approximation with Gaussian Kernels, SIAM Numer. Anal., 50 (2012), 247-271.  doi: 10.1137/10080138X.  Google Scholar

[14]

G. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using MATLAB, vol. 19 of Interdisciplinary mathematical Science, World Scientific Publishing, 2015. Google Scholar

[15]

W. W. Gao and Z. M. Wu, Quasi-interpolation for linear functional data, J. Comput. Appl. Math., 236 (2012), 3256-3264.  doi: 10.1016/j.cam.2012.02.028.  Google Scholar

[16]

W. Gao, X. Sun, Z. Wu and X. Zhou, Multivariate Monte Carlo approximation based on scattered data, SIAM J. Sci., (2020). Google Scholar

[17]

S. Hubbert, Closed form representations for a class of compactly supported radial basis functions, Adv. Comput. Math., 36 (2012), 115-136.  doi: 10.1007/s10444-011-9184-5.  Google Scholar

[18]

D. Husemöller, Elliptic Curves, Second edition, Graduate Texts in Mathematics, 111. Springer-Verlag, New York, 2004.  Google Scholar

[19]

V. Mazya and G. Schmidt, On approximate approximations using Gaussian kernels, IMA J. Numer. Anal., 16 (1996), 13-29.  doi: 10.1093/imanum/16.1.13.  Google Scholar

[20]

V. Mazya and G. Schmidt, Potentials of Gaussians and approximate wavelets, Math. Nachr., 280 (2007), 1176-1189.  doi: 10.1002/mana.200510544.  Google Scholar

[21]

M. J. D. Powell, The theory of radial basis function approximation in 1990, Advances in Numerical Analysis, Oxford Sci. Publ., Oxford Univ. Press, New York, 2 (1992), 105-210.   Google Scholar

[22]

W. Raji, A new proof of the transformation law of Jacobi's theta function $\theta_3(w, \tau)$, Proc. Amer. Math. Soc., 135 (2007), 3127-3132.  doi: 10.1090/S0002-9939-07-08867-3.  Google Scholar

[23]

S. D. Riemenschneider and N. Sivakumar, On cardinal interpolation by gaussian radial-basis functions: Properties of fundamental functions and estimates for lebesgue constants, J. Anal. Math., 79 (1999), 33-61.  doi: 10.1007/BF02788236.  Google Scholar

[24]

A. Ron, The L2-approximation orders of principal shift-invariant spaces generated by a radial basis function, in Numerical Methods in Approximation Theory, Vol. 105 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 9 (1992), 245-268. doi: 10.1007/978-3-0348-8619-2_14.  Google Scholar

[25]

R. Schaback, The missing Wendland functions, Adv. Comput. Math., 34 (2011), 67-81.  doi: 10.1007/s10444-009-9142-7.  Google Scholar

[26]

R. Schaback and H. Wendland, Inverse and saturation theorems for radial basis function interpolation, Math. Comp., 71 (2002), 669-681.  doi: 10.1090/S0025-5718-01-01383-7.  Google Scholar

[27]

S. Smale and D.-X. Zhou, Estimating the approximation error in learning theory, Anal. Appl. (Singap.), 1 (2003), 17-41.  doi: 10.1142/S0219530503000089.  Google Scholar

[28] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17. Cambridge University Press, Cambridge, 2005.   Google Scholar
[29] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511608759.  Google Scholar
[30]

Z. M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal., 13 (1993), 13-27.  doi: 10.1093/imanum/13.1.13.  Google Scholar

[31]

Q. Ye, Optimal designs of positive definite kernels for scattered data approximation, Appl. Comput. Harmon. Anal., 41 (2016), 214-236.  doi: 10.1016/j.acha.2015.08.009.  Google Scholar

[32]

Y. Ying and D.-X. Zhou, Learnability of Gaussians with flexible variances, J. Mach. Learn. Res., 8 (2007), 249-276.   Google Scholar

[33]

S. Zhu and A. J. Wathen, Convexity and solvability for compactly supported radial basis functions with different shapes, J. Sci. Comput., 63 (2015), 862-884.  doi: 10.1007/s10915-014-9919-9.  Google Scholar

[34]

M. V. Zhuravlev, E. A. Kiselev, L. A. Minin and S. M. Sitnik, Jacobi theta functions and systems of integer shifts of Gauss functions, J. Math. Sci. (N.Y.), 173 (2011), 231-241.  doi: 10.1007/s10958-011-0246-5.  Google Scholar

show all references

References:
[1]

B. J. C. Baxter, Norm estimates for inverses of Toeplitz distance matrices, J. Approx. Theory, 79 (1994), 222-242.  doi: 10.1006/jath.1994.1126.  Google Scholar

[2]

B. J. C. Baxter and N. Sivakumar, On shifted cardinal interpolation by Gaussians and multiquadrics, J. Approx. Theory, 87 (1996), 36-59.  doi: 10.1006/jath.1996.0091.  Google Scholar

[3]

R. Bellman, A Brief Introduction to Theta Functions, Athena Series: Selected Topics in Mathematics Holt, Rinehart and Winston, New York, 1961.  Google Scholar

[4]

J. P. Boyd, Error saturation in Gaussian radial basis functions on a finite interval, J. Comput. Appl. Math., 234 (2010), 1435-1441.  doi: 10.1016/j.cam.2010.02.019.  Google Scholar

[5]

J. P. Boyd and L. Wang, An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice, Appl. Math. Comput., 215 (2009), 2215-2223.  doi: 10.1016/j.amc.2009.08.037.  Google Scholar

[6]

M. D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Constr. Approx., 6 (1990), 225-255.  doi: 10.1007/BF01890410.  Google Scholar

[7] M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics, 12. Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511543241.  Google Scholar
[8]

A. Chernih and S. Hubbert, Closed form representations and properties of the generalised Wendland functions, J. Approx. Theory, 177 (2014), 17-33.  doi: 10.1016/j.jat.2013.09.005.  Google Scholar

[9]

Y. Choie and Y. Taguchi, A simple proof of the modular identity for theta series, Proc. Amer. Math. Soc., 133 (2005), 1935-1939.  doi: 10.1090/S0002-9939-05-07723-3.  Google Scholar

[10]

W. Couwenberg, A simple proof of the modular identity for theta functions, Proc. Amer. Math. Soc., 131 (2003), 3305-3307.  doi: 10.1090/S0002-9939-03-06902-8.  Google Scholar

[11]

T. A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions, Radial basis functions and partial differential equations, Comput. Math. Appl., 43 (2002), 413-422.  doi: 10.1016/S0898-1221(01)00295-4.  Google Scholar

[12]

G. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6437.  Google Scholar

[13]

G. Fasshauer, F. Hickernel and H. Woniakowski, On Dimension-independent Rates of Convergence for Function Approximation with Gaussian Kernels, SIAM Numer. Anal., 50 (2012), 247-271.  doi: 10.1137/10080138X.  Google Scholar

[14]

G. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using MATLAB, vol. 19 of Interdisciplinary mathematical Science, World Scientific Publishing, 2015. Google Scholar

[15]

W. W. Gao and Z. M. Wu, Quasi-interpolation for linear functional data, J. Comput. Appl. Math., 236 (2012), 3256-3264.  doi: 10.1016/j.cam.2012.02.028.  Google Scholar

[16]

W. Gao, X. Sun, Z. Wu and X. Zhou, Multivariate Monte Carlo approximation based on scattered data, SIAM J. Sci., (2020). Google Scholar

[17]

S. Hubbert, Closed form representations for a class of compactly supported radial basis functions, Adv. Comput. Math., 36 (2012), 115-136.  doi: 10.1007/s10444-011-9184-5.  Google Scholar

[18]

D. Husemöller, Elliptic Curves, Second edition, Graduate Texts in Mathematics, 111. Springer-Verlag, New York, 2004.  Google Scholar

[19]

V. Mazya and G. Schmidt, On approximate approximations using Gaussian kernels, IMA J. Numer. Anal., 16 (1996), 13-29.  doi: 10.1093/imanum/16.1.13.  Google Scholar

[20]

V. Mazya and G. Schmidt, Potentials of Gaussians and approximate wavelets, Math. Nachr., 280 (2007), 1176-1189.  doi: 10.1002/mana.200510544.  Google Scholar

[21]

M. J. D. Powell, The theory of radial basis function approximation in 1990, Advances in Numerical Analysis, Oxford Sci. Publ., Oxford Univ. Press, New York, 2 (1992), 105-210.   Google Scholar

[22]

W. Raji, A new proof of the transformation law of Jacobi's theta function $\theta_3(w, \tau)$, Proc. Amer. Math. Soc., 135 (2007), 3127-3132.  doi: 10.1090/S0002-9939-07-08867-3.  Google Scholar

[23]

S. D. Riemenschneider and N. Sivakumar, On cardinal interpolation by gaussian radial-basis functions: Properties of fundamental functions and estimates for lebesgue constants, J. Anal. Math., 79 (1999), 33-61.  doi: 10.1007/BF02788236.  Google Scholar

[24]

A. Ron, The L2-approximation orders of principal shift-invariant spaces generated by a radial basis function, in Numerical Methods in Approximation Theory, Vol. 105 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 9 (1992), 245-268. doi: 10.1007/978-3-0348-8619-2_14.  Google Scholar

[25]

R. Schaback, The missing Wendland functions, Adv. Comput. Math., 34 (2011), 67-81.  doi: 10.1007/s10444-009-9142-7.  Google Scholar

[26]

R. Schaback and H. Wendland, Inverse and saturation theorems for radial basis function interpolation, Math. Comp., 71 (2002), 669-681.  doi: 10.1090/S0025-5718-01-01383-7.  Google Scholar

[27]

S. Smale and D.-X. Zhou, Estimating the approximation error in learning theory, Anal. Appl. (Singap.), 1 (2003), 17-41.  doi: 10.1142/S0219530503000089.  Google Scholar

[28] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17. Cambridge University Press, Cambridge, 2005.   Google Scholar
[29] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511608759.  Google Scholar
[30]

Z. M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal., 13 (1993), 13-27.  doi: 10.1093/imanum/13.1.13.  Google Scholar

[31]

Q. Ye, Optimal designs of positive definite kernels for scattered data approximation, Appl. Comput. Harmon. Anal., 41 (2016), 214-236.  doi: 10.1016/j.acha.2015.08.009.  Google Scholar

[32]

Y. Ying and D.-X. Zhou, Learnability of Gaussians with flexible variances, J. Mach. Learn. Res., 8 (2007), 249-276.   Google Scholar

[33]

S. Zhu and A. J. Wathen, Convexity and solvability for compactly supported radial basis functions with different shapes, J. Sci. Comput., 63 (2015), 862-884.  doi: 10.1007/s10915-014-9919-9.  Google Scholar

[34]

M. V. Zhuravlev, E. A. Kiselev, L. A. Minin and S. M. Sitnik, Jacobi theta functions and systems of integer shifts of Gauss functions, J. Math. Sci. (N.Y.), 173 (2011), 231-241.  doi: 10.1007/s10958-011-0246-5.  Google Scholar

Figure 1.  $ \log_{10} \mathrm{csch} (\pi^2 d) $
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