Behaviour of the BMO-norm of the trigonometric Lagrange interpolation (Q2762957)
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scientific article; zbMATH DE number 1689808
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Behaviour of the BMO-norm of the trigonometric Lagrange interpolation |
scientific article; zbMATH DE number 1689808 |
Statements
16 January 2002
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Lagrange trigonometric interpolation polynomials
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BMO-norm
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0.8781283
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0.86792034
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0.8635603
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0.86339605
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Behaviour of the BMO-norm of the trigonometric Lagrange interpolation (English)
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A well-known fact is that there exists a continuous function on the interval \( [-1,1] \) such that the sequence of its Lagrange interpolation polynomials, with nodes \( x_{k}^{(n)} =\cos ((2k-1) \pi / 2n) \), \( k = 1,2,3, \ldots, n \), is everywhere divergent in this interval [\textit{J. Marcinkiewicz}, Acta Litt. Sci. Szeged 8, 131-135 (1937; Zbl 0016.10603)]. On the other hand if \( 1 < p < \infty \), then the sequence of the Lagrange trigonometric interpolation polynomials is convergent with respect to the \( L_{p} \)-norm. The author's main result (Satz 2) is that the last property is no longer valid for the BMO-norm.
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