Abstract
We consider certain aspects of the theory of interpolation via entire functions of exponential type, which include sampling node sequences χ which can be highly irregular and data sequences {y(x)}xn∈χ which are not necessarily bounded. Under appropriate conditions, we show that there is an entire function of a suitable exponential type which uniquely interpolates the data and indicate the validity of certain summabilty methods for the corresponding Lagrange type interpolation series. Some of our results significantly extend the work of Schoenberg,Cardinal interpolation and spline functions VII: The behavior of cardinal spline interpolation as their degree tends to infinity, J. Analyse Math.27 (1974), 205–229.
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References
P. J. Davis,Interpolation and Approximation, Dover, New York, 1975.
C. deBoor,A Practical Guide to Splines, Springer-Verlag, New York, 1978.
M. Golomb,H m,p-extensions byH m,p-extensions. J. Approx. Theory5 (1972), 238–275.
L Hörmander,Linear Partial Differential Operators, Springer-Verlag, New York, 1969.
S. V. Hrushchev, N. K. Nikol'skii and P. S. Pavlov,Unconditional bases of exponentials and of reproducing kernels, inComplex Analysis and Spectral Theory (V. P. Havin and N. K. Nikol'skii, eds.), Lecture Notes in Math.864, Springer-Verlag, Berlin, 1981, pp. 214–335.
M. J. Kadets,The exact valuer of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR155, No. 6 (1964), 1243–1254.
B. Ya. Levin,Lectures on Entire Functions, Translation of Mathematical Monographs, Vol. 150, Amer. Math. Soc., Providence, RI, 1996.
Yu. Lyubarskii and W. R. Madych,The recovery of irregularly sampled band limited functions via tempered splines, J. Funct. Anal.155 (1994), 201–222.
Yu. Lyubarskii and K. Seip,Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's (Ap) condition, Rev. Mat. Iberomericana13, No. 2 (1997), 361–376.
W. R. Madych,Spline type summability for multivariate sampling theory, inAnalysis of Divergence (W. O. Bray and C. V. Stanojević, eds.), Birkhäuser, Boston, 1999.
W. R. Madych and E. H. Potter,Error estimates for multivariate interpolation, J. Approx. Theory43 (1985), 132–139.
B. S. Pavlov,The basis property of a system of exponentials and the condition of Muckenhoupt, Dokl. Acad. Nauk SSSR247, no. 1 (1979), 37–40.
H. L. Royden,Real Analysis, 2nd ed., Macmillan Co., London, 1968.
I. J. Schoenberg,Cardinal interpolation and spline function VII: The behavior of cardinal spline interpolation as their degree tends to infinity, J. Analyse Math.27 (1974), 205–229.
E. T. Whitaker,On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburgh35 (1915), 181–194.
J. M. Whittaker,Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935.
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Madych, W.R. Summability of Lagrange type interpolation series. J. Anal. Math. 84, 207–229 (2001). https://doi.org/10.1007/BF02788110
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DOI: https://doi.org/10.1007/BF02788110