We obtain some upper and lower estimates for the sequences of the Lebesgue functions and constants of the Whittaker operatorsnx − kπ f kπ n for continuous functions. We give an analog of Nevai's formula for the Lagrange-Chebyshev and Lagrange-Laguerre interpolation polynomials for the operators under consideration. Its "local" version is established.

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We obtain asymptotic formulas for the values of second-order differential operators with the coefficient of bounded variation depending on the spectral parameter. We give a counterexample showing that the requirement of boundedness for the variation is essential for the preservation of the error of the so-obtained asymptotic formulas.

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We study approximation properties of various operators being the modifications of sinc approximations of continuous functions on an interval.

Abstract. Some properties of sinc-approximations of continuous functions on a segment are studied.For the first time, the sinc-approximations arose in the work of Pleynet. Later, in connection with developments in signals coding theory, E. Borel and E. T. Whittaker introduced the notions of a cardinal function and a truncated cardinal function, the restriction of which to [0, π] looks like this:(1)By the present time, the problem of sinc-approximation has been studied rather deeply for functions analytic on the real line and decaying exponentially at infinity (see, e.g.,The most complete survey of the results obtained in this direction before 1993, together with numerous important applications of sinc-approximations, can be found in [6]. An interesting historical survey of investigations on this topic is contained in [7].Also, a series of publications had their origin in what is called the sampling theorem or sometimes, the Whittaker-Kotel nikov-Shannon discretization theorem [8,9,6,11]; in those studies, various representations of entire functions were obtained in the form of series in sincs with interpolation nodes satisfying some "uniform distribution" conditions (see, e.g., [12]-[14]). Starting with the well-known paper [15] by Kramer, the relationship between the sampling theorems and the Lagrange interpolation with nodes in the SturmLiouville problem spectrum has been studied; see, e.g., [16].The sinc-approximations are widely applied for constructing numerical methods of mathematical physics and approximation theory for functions of one or several variables [17]-[21], in the theory of quadrature formulas [6,22] and wavelet transformations [8]-[11], [23,24]. In the paper [25], an analog of the sampling theorem that employs the Hermite type interpolation was obtained.The authors of [26] succeeded in proving approximability for less smooth functions. However, they were forced to modify the operator (1) somewhat. In [26], it was established that if a uniformly continuous and bounded functions f on R belongs to the Dini-Lipchitz class and satisfies the condition f (x) = O(|x| −δ ) as x → ±∞ for some 2010 Mathematics Subject Classification. Primary 42A10, 41A30.

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