c Allerton Press, Inc., 2016.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2016, Vol. 60, No. 3, pp. 63–71. c A.Yu. Trynin, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 3, pp. 72–81.
Original Russian Text Approximation of Continuous on a Segment Functions
With the Help of Linear Combinations of Sincs
A. Yu. Trynin1*
1
Saratov State University
ul. Astrakhanskaya 33, Saratov, 410012 Russia
Received July 11, 2014
Abstract—We investigate approximative properties of various operators, that are modifications of
sinc-approximations of continuous functions on a segment.
DOI: 10.3103/S1066369X16030087
Keywords: sinc-approximation, interpolation of functions, uniform approximation.
INTRODUCTION
In connection with the development of signal coding theory, E. Borel and E. T. Whittaker introduced
the notion of cardinal function and truncated cardinal function, whose restriction on the segment [0, π]
has the following form:
n
n
n
kπ
kπ
kπ
sin (nx − kπ)
(−1)k sin nx
f
=
f
=
.
(1)
lk,n (x)f
Ln (f, x) =
nx − kπ
n
nx − kπ
n
n
k=0
k=0
k=0
Sinc-approximations found a broad range of applications in the construction of various numerical
methods of mathematical physics and approximation theory of functions of both single and multiple
variables in theory of quadrature formulas in theory of wavelet transforms ([1], Chap. 7, § 4, item 2; [2]).
In [3], an estimate from above of the best approximation of continuous, vanishing at the endpoints
of the segment [0, π], functions by linear combinations of sincs was obtained. In [4, 5] estimates of the
error of uniform approximation on the entire axis by values of various operators, being combinations of
sincs, on the class of uniformly continuous and bounded on R functions were established. Note that
many operators considered in [4] in their construction are similar to the operators investigated in this
paper.
Unfortunately, in approximation of continuous functions on a segment with the help of (1) and many
other operators, in the neighborhood of the endpoints of the segment the Gribbs phenomenon arises (cf.,
e.g., [6]).
In [7] and [6] the authors obtained various estimates of the error of approximation of analytic in the
circle functions with the help of sinc-approximations (1). To the extent of my knowledge, before papers
[7] and [6], approximation by Whittaker cardinal functions on a segment, or a bounded interval was
carried out only for certain classes of analytic functions by reduction to the case of axis by means of
conformal transformation.
In [7], an analog of G. P. Nevai’s forumla, suitable for the investigation of operator (1), was obtained.
Papers [8, 9] are devoted to establishing necessary and sufficient conditions of pointwise and uniform,
inside interval (0, π), convergence of sinc-approximations (1) for continuous on [0, π] functions. The
authors of paper [10] use the results of paper [8] for the investigation of convergence of algorithms of
multi-level sinc-approximations of functions with minimal smoothness.
*
E-mail: [email protected].
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