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Approximation by optimal periodic interpolation. (English) Zbl 0743.41005

A periodic Hilbert space \(H_ d\) is defined by an \(\ell'\)-sequence \(d=(d_{-k})\) satisfying \(d_ 0=1\), \(d_{-k}=d_ k>0 (k\in\mathbb{N})\). \(H_ d:=\left\{f\in L^ 2_{2\pi}:\sum^{\alpha}_{- \infty}|(f,e_ k)|^ 2/d_ k<\infty\right\}\) is the Hilbert space of continuous \(2\pi\)-periodic functions with inner product \((f,g)_ d=\sum^{\infty}_{k=-\infty}(f,e_ k)(e_ k,g)/d_ k\), \(e_ k=\exp(ikt)\), \(k\in\mathbb{Z}\) where \((f,g)={1\over 2\pi}\int^{2\pi}_ 0 f(t)\overline{g(t)}dt\).
The author studies the approximation power of optimal periodic interpolation in the uniform norm, a subject studied by M. Prager [ibid. 24, 406-420 (1979; Zbl 0449.41003)] and later by the author in 1985 and 1987. For given \(f\in H_ d\), \(S_ n(f)\) is the unique interpolant of \(f\) in \(H_ d\) with minimum norm, i.e., \(S_ n(f)(t_ j)=f(t_ j)\), \((0\leq j<n)\), \(\| S_ n(f)\|_ d=\min\{\| g\|_ d: g(t_ j)=f(t_ j)\), \(0\leq j<n\)}. Let \(n=2m+1\) and \(D_ n:=\sum^{\infty}_{r=1}d_{rm}\). Let \(A\) be the linear operator in \(L^ 2_{2\pi}\) given by \(Af=\sum^{\infty}_{k=-\infty}(f,e_ k)d^{-1}_ ke_ k\). Among the results he proves the following estimate: \(\| f-S_ n(f)\|_{\infty}\leq 2D_ n\| Af\|_ a\), where \(\| Af\|_ a=\sum^{\infty}_{k=-\infty}\|(f,e_ k)\|\).

MSC:

41A05 Interpolation in approximation theory
41A30 Approximation by other special function classes

Citations:

Zbl 0449.41003
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References:

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[2] F.-J. Delvos: Periodic interpolation on uniform meshes. J. Approx. Theory 51 (1987), 71 - 80. · Zbl 0664.41004 · doi:10.1016/0021-9045(87)90096-7
[3] F.-J. Delvos: Interpolation of odd periodic functions on uniform meshes. ISNM 74 (1985), 105-121.. · Zbl 0578.41012
[4] Y. Katznelson: An introduction to harmonic analysis. Dover, New York 1976. · Zbl 0352.43001
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[6] M. Práger: Universally optimal approximation of functionals. Apl. mat. 24 (1979), 406-420. · Zbl 0449.41003
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