Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration
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Numer. Math. (2013) 124:361–394 DOI 10.1007/s00211-013-0515-y
Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration Tomoaki Okayama · Takayasu Matsuo · Masaaki Sugihara
Received: 14 January 2009 / Revised: 19 October 2012 / Published online: 20 March 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given. Mathematics Subject Classification (1991)
41A05 · 41A55 · 65D05 · 65D30
T. Okayama · T. Matsuo · M. Sugihara Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Present Address: T. Okayama (B) Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan e-mail: [email protected]
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1 Introduction The Sinc approximation on the whole real line is expressed as n F( j h)S( j, h)(x), x ∈ R, F(x) ≈
(1.1)
j=−n
where S( j, h)(x) is the so-called Sinc function defined by sin[π(x/ h − j)] S( j, h)(x) = , π(x/ h − j) and h is a mesh size appropriately selected depending on n. A variety of approximation formulas are derived from the Sinc approximation. For example, the Sinc quadrature for the integral on (−∞, ∞) is derived by integrating both sides of (1.1): ∞ F(x) dx ≈ −∞
⎧ ∞ ⎨ n −∞
⎩
⎫ ⎬ F( j h)S( j, h)(x)
⎭
j=−n
dx = h
n
F( j h),
(1.2)
j=−n
which
∞ coincides with the standard (truncated) trapezoidal formula. Here the relation −∞ S( j, h)(x) dx = h is used. Another example is the Sinc indefinite integration expressed as ⎫ ⎧ x x ⎨ n n ⎬ F(σ ) dσ ≈ F( j h)S( j, h)(σ ) dσ = F( j h)J ( j, h)(x), (1.3) ⎭ ⎩ −∞
−∞
j=−n
j=−n
the basis function J ( j, h)(x) is computed via the sine integral function Si(x) =
where x {sin(σ )/σ } dσ : 0 J ( j, h)(x) = h
1 1 + Si[π(x/ h − j)] . 2 π
(1.4)
Other examples include Sinc indefinite convolution, Harmonic-Sinc approximation, approximation of derivatives, approximation of Hilbert and Cauchy transforms, and approximation of inversion of Fourier and Laplace transforms (see, for example, Stenger [18,19]). When the target interval is finite,
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