Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

Muradyan, O. A.; Khavinson, S. Ya.

doi:10.1007/bf01780974

Cited by 7 publications

(6 citation statements)

References 2 publications

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“…We restrict ourselves by a simple statement: main asymptotic formula (16) contains certain subtleties related to the parity of 8. Nevertheless, formulae (16)-(18) describe well the growth of the maximal coefficient in studied Bernstein polynomials (7). The growth of such rate turns out to be exponential of order 2 = 2 /2 .…”

Section: Proof Of Theoremmentioning

confidence: 89%

“…But the real picture becomes more complicated: the main "strategic" tendency in the behavior of 2 becomes evident only for sufficiently large incides = 2 . It is quite problematic to find out such tendency basing on an initial definition (5) and not using explicit writing (7). This is a principal difference of the situation with the function ( ) = | | on [−1, 1] from a similar example ( ) = |2 −1| on [0, 1], where a swift growth of the coefficients is evidently seen while opening the brackets already at the first indices of Bernstein polynomials (see [8]).…”

Section: Formulation Of Problemmentioning

confidence: 99%

“…Theorem 2. Let the quantity 2 be defined by formula (12) as the maximal absolute value of the coefficients (9) taken from polynomials (7). Then the asymptotic formula holds:…”

Section: Main Statementsmentioning

confidence: 99%

“…Relations (16)- (18) show that the maximal (by the absolute value) coefficient in polynomials (7) grows with at most exponential rate but this rate is essentially small than in the analogous example on the standard segment [0, 1] (cf. [8], [9], [15]).…”

Section: Main Statementsmentioning

confidence: 99%

“…There is one special direction in the approximation theory devoted to studying possible growth rate of the coefficients in algebraic polynomials used in uniform approximations of continuous functions on a given set [1]- [7]. Some results, see [3]- [6], were obtained by estimating the coefficients in the corresponding Bernstein polynomials.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On growth rate of coefficients in Bernstein polynomials for the standard modulus function on a symmetric interval

Petrosova¹,

Tikhonov²,

Sherstyukov³

2018

Ufimskii Matematicheskii Zhurnal

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The subject of the paper is closely related to one general direction in the approximation theory, within which the growth rate of the coefficients of algebraic polynomials is studied for uniform approximations of continuous functions. The classical Bernstein polynomials play an important role here. We study in detail a model example of Bernstein polynomials for the standard modulus function on a symmetric interval. The question under consideration is the growth rate of of the coefficients in these polynomials with an explicit algebraic representation. It turns out that in the first fifteen polynomials the growth of the coefficients is almost not observed. For the next polynomials the situation changes, and coefficients of exponential growth appear. Our main attention is focused on the behaviour of the maximal coefficient, for which exact exponential asymptotics and corresponding two-sided estimates are established (see Theorem 2). As it follows from the obtained result, the maximal coefficient has growth 2 /2 / 2 , where is the index of the Bernstein polynomial. It is shown that the coefficients equidistant from the maximal one have a similar growth rate (for details, see Theorem 3). The group of the largest coefficients is located in the central part of the Bernstein polynomials but at the ends the coefficients are sufficiently small. The behavior of the sum of absolute values of all coefficients is also considered. This sum admits an explicit expression that is not computable in the sense of traditional combinatorial identities. On the base of a preliminary recurrence relation, we succeeded to obtain the exact asymptotics for the sum of absolute values of all coefficients and to give the corresponding two-sided estimates (see Theorem 4). The growth rate of the sum is 2 /2 / 3/2. In the end of the paper, we compare this result with a general Roulier estimate and new related problems are formulated.

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Section: Proof Of Theoremmentioning

confidence: 89%

Section: Formulation Of Problemmentioning

confidence: 99%

“…Theorem 2. Let the quantity 2 be defined by formula (12) as the maximal absolute value of the coefficients (9) taken from polynomials (7). Then the asymptotic formula holds:…”

Section: Main Statementsmentioning

confidence: 99%

Section: Main Statementsmentioning

confidence: 99%