# American Institute of Mathematical Sciences

July  2021, 14(7): 2399-2416. doi: 10.3934/dcdss.2020403

## Interpolation of exponential-type functions on a uniform grid by shifts of a basis function

 1 Department of Mathematics, University of Leicester, Leicester, UK 2 Department of Mathematics, Missouri State University, USA 3 Department of Mathematics, Çankaya University, Ankara, Turkey

* Corresponding author: Alexander Kushpel

Dedicated to the memory of Ward Cheney

Received  December 2019 Published  July 2021 Early access  June 2020

Fund Project: The first author was supported by EPSRC Grant EP/H020071/1, the University of Leicester via study and Missouri State University through a generous travel grant

In this paper, we present a new approach to solving the problem of interpolating a continuous function at $(n+1)$ equally-spaced points in the interval $[0, 1]$, using shifts of a kernel on the $(1/n)$-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Finally we give a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval (assuming knowledge of the discrete moments of the Gaussian).

Citation: Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403
##### References:
 [1] M. Abramowitz and I. A. Stegun, Orthogonal Polynomials, in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972,771–802. [2] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. [3] 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. [4] C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx., 3 (1987), 199-208.  doi: 10.1007/BF01890564. [5] E. Carneiro, F. Littmann and J. D. Vaaler, Gaussian subordination for the Beurling-Selberg extremal problem, Trans. Amer. Math. Soc., 365 (2013), 3493-3534.  doi: 10.1090/S0002-9947-2013-05716-9. [6] E. Carneiro and J. D. Vaaler, Some extremal functions in Fourier analysis. II, Trans. Amer. Math. Soc., 362 (2010), 5803-5843.  doi: 10.1090/S0002-9947-2010-04886-X. [7] D. Chen and W. Cheney, Lagrange polynomial interpolation, in Approximation Theory XII, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2008, 60–76. [8] E. W. Cheney and X. Sun, Interpolation on spheres by positive definite functions, in Approximation Theory, Monogr. Textbooks Pure Appl. Math., 212, Dekker, New York, 1998,141–156. [9] W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl., 52/53 (1983), 217-234.  doi: 10.1016/0024-3795(83)80015-9. [10] G. Fix and G. Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory, Studies in Appl. Math., 48 (1969), 265-273.  doi: 10.1002/sapm1969483265. [11] B. Fornberg, E. Larsson and N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33 (2011), 869-892.  doi: 10.1137/09076756X. [12] E. H. Georgoulis, J. Levesley and F. Subhan, Multilevel sparse kernel-based interpolation, SIAM J. Sci. Comput., 35 (2013), 815-831.  doi: 10.1137/110859610. [13] S. M. Gomes, A. K. Kushpel, J. Levesley and D. L. Ragozin, Interpolation on the torus using sk-splines with number-theoretic knots, J. Approx. Theory, 98 (1999), 56-71.  doi: 10.1006/jath.1998.3278. [14] M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, North-Holland, Amsterdam, 1992,263–281. [15] K. Hamm, Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions, J. Approx. Theory, 189 (2015), 101-122.  doi: 10.1016/j.jat.2014.10.011. [16] T. Hangelbroek, W. Madych, F. Narcowich and J. D. Ward, Cardinal interpolation with Gaussian kernels, J. Fourier Anal. Appl., 18 (2012), 67-86.  doi: 10.1007/s00041-011-9185-2. [17] S. Hubbert and J. Levesley, Convergence of multilevel stationary Gaussian convolution, in Numerical Mathematics and Advanced Applications, Lect. Notes Comput. Sci. Eng., 126, Springer, Cham, 2019, 83–92. doi: 10.1007/978-3-319-96415-7_5. [18] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. [19] A. K. Kushpel, Sharp estimates for the widths of convolution classes, Math. USSR-Izv., 33 (1989), 631-649.  doi: 10.1070/IM1989v033n03ABEH000862. [20] A. K. Kushpel, Estimates of the diameters convolution classes in the spaces C and L, Ukrain. Math. J., 41 (1989), 919-924.  doi: 10.1007/BF01058308. [21] A. K. Kushpel, Convergence of $sk$-splines in $L_q$. I, Int. J. Pure Appl. Math., 45 (2008), 87-101. [22] A. K. Kushpel, Convergence of $sk$-splines in $L_q$. II, Int. J. Pure Appl. Math., 45 (2008), 103-119. [23] A. K. Kushpel, A method of inversion of Fourier transforms and its applications, Int. J. of Diff. Equations Appl., 18 (2019), 25-29. [24] J. Levesley and A. K. Kushpel, Generalised $sk$-spline interpolation on compact abelian groups, J. Approx. Theory, 97 (1999), 311-333.  doi: 10.1006/jath.1997.3267. [25] W. A. Light and E. W. Cheney, Interpolation by periodic radial basis functions, J. Math. Anal. Appl., 168 (1992), 111-130.  doi: 10.1016/0022-247X(92)90193-H. [26] W. R. Madych and S. A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. Approx. Theory, 70 (1992), 94-114.  doi: 10.1016/0021-9045(92)90058-V. [27] F. J. Narcowich, J. D. Ward and H. Wendland, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constr. Approx., 24 (2006), 175-186.  doi: 10.1007/s00365-005-0624-7. [28] F. J. Narcowich, X. Sun, J. D. Ward and H. Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math., 7 (2007), 369-390.  doi: 10.1007/s10208-005-0197-7. [29] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. [30] 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. [31] S. D. Riemenschneider and N. Sivakumar, Cardinal interpolation by Gaussian functions: A survey, J. Anal., 8 (2000), 157-178. [32] A. Ron, A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution, Constr. Approx., 5 (1989), 297-308.  doi: 10.1007/BF01889611. [33] A. Ron, Introduction to shift-invariant spaces. Linear independence, in Multivariate Approximation and Applications, Cambridge Univ. Press, Cambridge, 2001, 112–151. doi: 10.1017/CBO9780511569616.006. [34] L. A. Rubel, Necessary and sufficient conditions for Carlson's theorem on entire functions, Trans. Amer. Math. Soc., 83 (1956), 417-429.  doi: 10.2307/1992882. [35] I. J. Schoenberg, Cardinal Spline Interpolation, Conference Board of the Mathematical Sciences Regional Regional Conference Series in Applied Mathematics, 12, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. doi: 10.1137/1.9781611970555. [36] C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379–423,623–656. doi: 10.1002/j.1538-7305.1948.tb01338.x. [37] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971. doi: 10.1515/9781400883899. [38] Y. Xu and E. W. Cheney, Interpolation by periodic radial functions. Advances in the theory and applications of radial basis functions, Comput. Math. Appl., 24 (1992), 201-215.  doi: 10.1016/0898-1221(92)90181-G. [39] K. Yosida, Functional Analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York; Springer-Verlag, Berlin, 1965. doi: 10.1007/978-3-662-25762-3. [40] A. I. Zayed, Advances in Shannon’s Sampling Theory, CRC Press, Boca Raton, FL, 1993.

show all references

##### References:
 [1] M. Abramowitz and I. A. Stegun, Orthogonal Polynomials, in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972,771–802. [2] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. [3] 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. [4] C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx., 3 (1987), 199-208.  doi: 10.1007/BF01890564. [5] E. Carneiro, F. Littmann and J. D. Vaaler, Gaussian subordination for the Beurling-Selberg extremal problem, Trans. Amer. Math. Soc., 365 (2013), 3493-3534.  doi: 10.1090/S0002-9947-2013-05716-9. [6] E. Carneiro and J. D. Vaaler, Some extremal functions in Fourier analysis. II, Trans. Amer. Math. Soc., 362 (2010), 5803-5843.  doi: 10.1090/S0002-9947-2010-04886-X. [7] D. Chen and W. Cheney, Lagrange polynomial interpolation, in Approximation Theory XII, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2008, 60–76. [8] E. W. Cheney and X. Sun, Interpolation on spheres by positive definite functions, in Approximation Theory, Monogr. Textbooks Pure Appl. Math., 212, Dekker, New York, 1998,141–156. [9] W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl., 52/53 (1983), 217-234.  doi: 10.1016/0024-3795(83)80015-9. [10] G. Fix and G. Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory, Studies in Appl. Math., 48 (1969), 265-273.  doi: 10.1002/sapm1969483265. [11] B. Fornberg, E. Larsson and N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33 (2011), 869-892.  doi: 10.1137/09076756X. [12] E. H. Georgoulis, J. Levesley and F. Subhan, Multilevel sparse kernel-based interpolation, SIAM J. Sci. Comput., 35 (2013), 815-831.  doi: 10.1137/110859610. [13] S. M. Gomes, A. K. Kushpel, J. Levesley and D. L. Ragozin, Interpolation on the torus using sk-splines with number-theoretic knots, J. Approx. Theory, 98 (1999), 56-71.  doi: 10.1006/jath.1998.3278. [14] M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, North-Holland, Amsterdam, 1992,263–281. [15] K. Hamm, Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions, J. Approx. Theory, 189 (2015), 101-122.  doi: 10.1016/j.jat.2014.10.011. [16] T. Hangelbroek, W. Madych, F. Narcowich and J. D. Ward, Cardinal interpolation with Gaussian kernels, J. Fourier Anal. Appl., 18 (2012), 67-86.  doi: 10.1007/s00041-011-9185-2. [17] S. Hubbert and J. Levesley, Convergence of multilevel stationary Gaussian convolution, in Numerical Mathematics and Advanced Applications, Lect. Notes Comput. Sci. Eng., 126, Springer, Cham, 2019, 83–92. doi: 10.1007/978-3-319-96415-7_5. [18] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. [19] A. K. Kushpel, Sharp estimates for the widths of convolution classes, Math. USSR-Izv., 33 (1989), 631-649.  doi: 10.1070/IM1989v033n03ABEH000862. [20] A. K. Kushpel, Estimates of the diameters convolution classes in the spaces C and L, Ukrain. Math. J., 41 (1989), 919-924.  doi: 10.1007/BF01058308. [21] A. K. Kushpel, Convergence of $sk$-splines in $L_q$. I, Int. J. Pure Appl. Math., 45 (2008), 87-101. [22] A. K. Kushpel, Convergence of $sk$-splines in $L_q$. II, Int. J. Pure Appl. Math., 45 (2008), 103-119. [23] A. K. Kushpel, A method of inversion of Fourier transforms and its applications, Int. J. of Diff. Equations Appl., 18 (2019), 25-29. [24] J. Levesley and A. K. Kushpel, Generalised $sk$-spline interpolation on compact abelian groups, J. Approx. Theory, 97 (1999), 311-333.  doi: 10.1006/jath.1997.3267. [25] W. A. Light and E. W. Cheney, Interpolation by periodic radial basis functions, J. Math. Anal. Appl., 168 (1992), 111-130.  doi: 10.1016/0022-247X(92)90193-H. [26] W. R. Madych and S. A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. Approx. Theory, 70 (1992), 94-114.  doi: 10.1016/0021-9045(92)90058-V. [27] F. J. Narcowich, J. D. Ward and H. Wendland, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constr. Approx., 24 (2006), 175-186.  doi: 10.1007/s00365-005-0624-7. [28] F. J. Narcowich, X. Sun, J. D. Ward and H. Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math., 7 (2007), 369-390.  doi: 10.1007/s10208-005-0197-7. [29] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. [30] 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. [31] S. D. Riemenschneider and N. Sivakumar, Cardinal interpolation by Gaussian functions: A survey, J. Anal., 8 (2000), 157-178. [32] A. Ron, A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution, Constr. Approx., 5 (1989), 297-308.  doi: 10.1007/BF01889611. [33] A. Ron, Introduction to shift-invariant spaces. Linear independence, in Multivariate Approximation and Applications, Cambridge Univ. Press, Cambridge, 2001, 112–151. doi: 10.1017/CBO9780511569616.006. [34] L. A. Rubel, Necessary and sufficient conditions for Carlson's theorem on entire functions, Trans. Amer. Math. Soc., 83 (1956), 417-429.  doi: 10.2307/1992882. [35] I. J. Schoenberg, Cardinal Spline Interpolation, Conference Board of the Mathematical Sciences Regional Regional Conference Series in Applied Mathematics, 12, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. doi: 10.1137/1.9781611970555. [36] C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379–423,623–656. doi: 10.1002/j.1538-7305.1948.tb01338.x. [37] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971. doi: 10.1515/9781400883899. [38] Y. Xu and E. W. Cheney, Interpolation by periodic radial functions. Advances in the theory and applications of radial basis functions, Comput. Math. Appl., 24 (1992), 201-215.  doi: 10.1016/0898-1221(92)90181-G. [39] K. Yosida, Functional Analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York; Springer-Verlag, Berlin, 1965. doi: 10.1007/978-3-662-25762-3. [40] A. I. Zayed, Advances in Shannon’s Sampling Theory, CRC Press, Boca Raton, FL, 1993.
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