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Norms on L of Periodic Interpolation Splines with Equidistant Nodes

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Abstract

We consider the set S r,n of periodic (with period 1) splines of degree r with deficiency 1 whose nodes are at n equidistant points x i=i / n. For n-tuples y = (y 0, ... , y n-1), we take splines s r,n (y, x) from S r,n solving the interpolation problem

$$s_{r,n} (y,t_i ) = y_i,$$

where t i = x i if r is odd and t i is the middle of the closed interval [x i , x i+1 ] if r is even. For the norms L * r,n of the operator ys r,n (y, x) treated as an operator from l 1 to L 1 [0, 1] we establish the estimate

$$L_{r,n}^ * = \frac{4}{{\pi ^2 n}}log min(r,n) + O\left( {\frac{1}{n}} \right)$$

with an absolute constant in the remainder. We study the relationship between the norms L * r,n and the norms of similar operators for nonperiodic splines.

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REFERENCES

  1. Yu. N. Subbotin and S. A. Telyakovskii, “Asymptotics of Lebesgue constants of periodic interpolation splines with equidistant nodes,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 191 (2000), no. 8, 131–140.

    Google Scholar 

  2. S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Fizmatgiz, Moscow, 1976.

    Google Scholar 

  3. A. A. Zhensykbaev, “Sharp estimates of the uniform approximation of continuous periodic functions by splines of rth order” Mat. Zametki [Math. Notes], 13 (1973), no. 2, 217–228.

    Google Scholar 

  4. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow, 1962; English transl.: Academic Press, New York–London, 1965.

    Google Scholar 

  5. M. J. Marsden and F. B. Richards, and S. D. Riemenschneider, “Cardinal spline interpolation operators on l p data,” Indiana University Math. J., 24 (1975), 677–689.

    Google Scholar 

  6. I. J. Schoenberg, “Cardinal interpolation and spline functions,” J. Approx. Theory, 2 (1969), 167–206.

    Google Scholar 

  7. Yu. N. Subbotin, “On the relation between finite differences and corresponding derivatives,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 78 (1965), 24–42.

    Google Scholar 

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Subbotin, Y.N., Telyakovskii, S.A. Norms on L of Periodic Interpolation Splines with Equidistant Nodes. Mathematical Notes 74, 100–109 (2003). https://doi.org/10.1023/A:1025075301686

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  • DOI: https://doi.org/10.1023/A:1025075301686

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