Remark on the degree of approximation of continuous functions by singular integrals. (English) Zbl 0808.42004
Let \(f\) be a bounded, real-valued function defined on the real line \(\mathbb{R}\) or \([-\pi,\pi]\). R. N. Mohapatra and R. S. Rodriguez [Math. Nachr. 149, 117-124 (1990; Zbl 0726.42003)] obtained the rate of convergence of singular integrals of \(f\) to \(f\) in the Hölder metric under the assumption:
\[
t^{-1} \omega(f;t)\quad\text{is decreasing with }t>0,{(*)}
\]
where \(\omega(f;t)\) is the modulus of continuity of \(f\).
In the present paper, the author has shown that for \(f\in C_{2\pi}\), the estimates obtained by Mohapatra and Rodriguez (ibid.) may be obtained without using \((*)\).
Remark: The author has mentioned that Mohapatra and Rodriguez (ibid.) has obtained the rate of convergence for \(f\in C_{2\pi}\), which is not correct. In fact, these authors did not use the periodicity of \(f\).
In the present paper, the author has shown that for \(f\in C_{2\pi}\), the estimates obtained by Mohapatra and Rodriguez (ibid.) may be obtained without using \((*)\).
Remark: The author has mentioned that Mohapatra and Rodriguez (ibid.) has obtained the rate of convergence for \(f\in C_{2\pi}\), which is not correct. In fact, these authors did not use the periodicity of \(f\).
Reviewer: P.Chandra (Ujjain)
MSC:
42A10 | Trigonometric approximation |
42A50 | Conjugate functions, conjugate series, singular integrals |
41A25 | Rate of convergence, degree of approximation |
41A35 | Approximation by operators (in particular, by integral operators) |
Keywords:
Picard singular integral; Poisson-Cauchy singular integral; Gauss- Weierstraß singular integral; degree of approximation; approximation by integral operators; rate of convergenceCitations:
Zbl 0726.42003References:
[1] | Approximation of functions, Holt, Rinehart and Winston, New York, 1986 |
[2] | Mohapatra, Math. Nachr. 149 pp 117– (1990) · Zbl 0726.42003 · doi:10.1002/mana.19901490108 |
[3] | Peetre, Ricerche Mat. 18 pp 239– (1969) |
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