Abstract:
The result obtained in this paper allows one to identify the approximate convergence at a point (or its absence) of the values of the Whittaker operators:
$$
L_n(f,x)=\sum_{k=0}^{n}\frac{\sin(nx-k\pi)}{nx-k\pi}\,f\biggl(\frac{k\pi}{n}\biggr).
$$
The only requirement on the function $f$ to be approximated is its continuity on $[0,\pi]$. The information about $f$ can be reduced to its values at the nodes $k\pi/n$ lying in a neighbourhood of the point at which the approximation properties are actually under
consideration.
A test for the uniform convergence of these operators on compact subsets of $(0,\pi)$ is also obtained for continuous functions, which is similar to Privalov's criterion of the convergence of the Lagrange–Chebyshev interpolation polynomials and trigonometric polynomials.
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