Dedicated to Professor Giuseppe Mastroianni on occasion of his 70th birthday Abstract. There are ... more Dedicated to Professor Giuseppe Mastroianni on occasion of his 70th birthday Abstract. There are two kinds of estimates of the degree of approximation of continuous functions on [−1, 1] by algebraic polynomials, Nikolskii-type pointwise estimates and Jackson-type uniform estimates, involving either ordinary moduli of smoothness, or the Ditzian-Totik (DT) ones, or the recent estimates involving the weighted DTmoduli of smoothness. The purpose of this paper is to complete the table of the validity or invalidity of the pointwise estimates in the case of coconvex polynomial approximation. This will enable us to compare this table with a corresponding one for coconvex Jackson-type (uniform) estimates.
ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Pre... more ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
We survey developments, over the last thirty years, in the theory of Shape Preserving Approximati... more We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a
For natural k and n ≥ 2k, we determine the exact constant c(n, k) in Dzyadyk’s inequality Pn′φn1−... more For natural k and n ≥ 2k, we determine the exact constant c(n, k) in Dzyadyk’s inequality Pn′φn1−kC−11≤cnknPnφn−kC−11$$ {\left\Vert {P}_n^{\prime }{\varphi}_n^{1-k}\right\Vert}_{C\left[-1,1\right]}\le c\left(n,k\right)n{\left\Vert {P}_n{\varphi}_n^{-k}\right\Vert}_{C\left[-1,1\right]} $$for the derivativePn′$$ {P}_n^{\prime } $$of an algebraic polynomialPnof degree≤n,whereφnx≔n−2+1−x2.$$ {\varphi}_n(x):= \sqrt{n^{-2}+1-{x}^2}. $$Namely,cnk=1+k1+n2−1n2−k.$$ c\left(n,k\right)={\left(1+k\frac{\sqrt{1+{n}^2}-1}{n}\right)}^2-k. $$
Let be an arbitrary function of the type of a second modulus of continuity. It is proved that if ... more Let be an arbitrary function of the type of a second modulus of continuity. It is proved that if is a given function, and (1)for any triple of points and , then this function is the trace of some continuous function for which , where is an absolute constant. The function is constructed by a formula which uses only the values of on and the values of . The converse of this assertion, namely, that condition (1) holds for each continuous function on any set , can be verified without difficulty. Bibliography: 7 titles.
In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, wh... more In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y s = {y i } s i=1 of points y i ∈ (-1, 1), $$ \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_s}} \right):n \geqslant \mathcal{N}*} \right\} \leqslant c\left( {\alpha, s} \right)\sup \left\{ {{n^\alpha }{E_n}(f):n \geqslant 1} \right\}, $$ where E n (f) and E (2) n (f, Y s ) denote, respectively, the degrees of the best unconstrained and (co)convex approximations and c(α, s) is a constant depending only on α and s. Moreover, it has been shown that $$ \mathcal{N}* $$ may be chosen to be 1 for s = 0 or s = 1, α ≠ 4, and that it must depend on Y s and α for s = 1, α = 4 or s ≥ 2. In Part II of the paper, we show that a more general inequality $$ \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_s}} \right):n \geqslant \mathcal{N}*} \right\} \leqslant c\left( {\alpha, \mathcal{N},s} \right)\sup \left\{ {{n^\alpha }{E_n}(f)...
ABSTRACT points ~a and ~b" We assume the curve F to be given in the parametric form: r= ... more ABSTRACT points ~a and ~b" We assume the curve F to be given in the parametric form: r= {~:~= ~(s),sES}, (1) where s is the arc-abscissa (i.e.,length of arc, with approprate sign, from a fixed point on P), the :positive direction of which corresponds to motion of the point ~E F from point ~a to point ~b. If, however, ~a = [b, i~ if the curve F is closed, we take the positive direction to be that corresponding to motion of the point ~E F so that the domain D bounded by the curve F will stay always to the left. Moreover, for the point ~a df df = ~b we taken an arbitrary point on the curve F, not a corner point, and we put [0 = ~a, ~k+l = ~b' The domain S for the variation of the arc-abscissa s for the case in which ~a ~ ~b is an interval in, b]; if ~a = ~b, we take the set S to be the whole real axis and the function ~(s) to be periodic with period 2l. For arbitrary e, dES we denoted by Fie, d] the arc of the curve F having endpoints at ~(c) and ~(d), ioeo P[c, dl = {~ : ; = ~ (s), s E [c, dl}. (2)
ABSTRACT For every n, we compute the Lebesgue constant of Rogosinski kernel with any preassigned ... more ABSTRACT For every n, we compute the Lebesgue constant of Rogosinski kernel with any preassigned accuracy.
Dedicated to Professor Giuseppe Mastroianni on occasion of his 70th birthday Abstract. There are ... more Dedicated to Professor Giuseppe Mastroianni on occasion of his 70th birthday Abstract. There are two kinds of estimates of the degree of approximation of continuous functions on [−1, 1] by algebraic polynomials, Nikolskii-type pointwise estimates and Jackson-type uniform estimates, involving either ordinary moduli of smoothness, or the Ditzian-Totik (DT) ones, or the recent estimates involving the weighted DTmoduli of smoothness. The purpose of this paper is to complete the table of the validity or invalidity of the pointwise estimates in the case of coconvex polynomial approximation. This will enable us to compare this table with a corresponding one for coconvex Jackson-type (uniform) estimates.
ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Pre... more ABSTRACT Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
We survey developments, over the last thirty years, in the theory of Shape Preserving Approximati... more We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a
For natural k and n ≥ 2k, we determine the exact constant c(n, k) in Dzyadyk’s inequality Pn′φn1−... more For natural k and n ≥ 2k, we determine the exact constant c(n, k) in Dzyadyk’s inequality Pn′φn1−kC−11≤cnknPnφn−kC−11$$ {\left\Vert {P}_n^{\prime }{\varphi}_n^{1-k}\right\Vert}_{C\left[-1,1\right]}\le c\left(n,k\right)n{\left\Vert {P}_n{\varphi}_n^{-k}\right\Vert}_{C\left[-1,1\right]} $$for the derivativePn′$$ {P}_n^{\prime } $$of an algebraic polynomialPnof degree≤n,whereφnx≔n−2+1−x2.$$ {\varphi}_n(x):= \sqrt{n^{-2}+1-{x}^2}. $$Namely,cnk=1+k1+n2−1n2−k.$$ c\left(n,k\right)={\left(1+k\frac{\sqrt{1+{n}^2}-1}{n}\right)}^2-k. $$
Let be an arbitrary function of the type of a second modulus of continuity. It is proved that if ... more Let be an arbitrary function of the type of a second modulus of continuity. It is proved that if is a given function, and (1)for any triple of points and , then this function is the trace of some continuous function for which , where is an absolute constant. The function is constructed by a formula which uses only the values of on and the values of . The converse of this assertion, namely, that condition (1) holds for each continuous function on any set , can be verified without difficulty. Bibliography: 7 titles.
In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, wh... more In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y s = {y i } s i=1 of points y i ∈ (-1, 1), $$ \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_s}} \right):n \geqslant \mathcal{N}*} \right\} \leqslant c\left( {\alpha, s} \right)\sup \left\{ {{n^\alpha }{E_n}(f):n \geqslant 1} \right\}, $$ where E n (f) and E (2) n (f, Y s ) denote, respectively, the degrees of the best unconstrained and (co)convex approximations and c(α, s) is a constant depending only on α and s. Moreover, it has been shown that $$ \mathcal{N}* $$ may be chosen to be 1 for s = 0 or s = 1, α ≠ 4, and that it must depend on Y s and α for s = 1, α = 4 or s ≥ 2. In Part II of the paper, we show that a more general inequality $$ \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_s}} \right):n \geqslant \mathcal{N}*} \right\} \leqslant c\left( {\alpha, \mathcal{N},s} \right)\sup \left\{ {{n^\alpha }{E_n}(f)...
ABSTRACT points ~a and ~b" We assume the curve F to be given in the parametric form: r= ... more ABSTRACT points ~a and ~b" We assume the curve F to be given in the parametric form: r= {~:~= ~(s),sES}, (1) where s is the arc-abscissa (i.e.,length of arc, with approprate sign, from a fixed point on P), the :positive direction of which corresponds to motion of the point ~E F from point ~a to point ~b. If, however, ~a = [b, i~ if the curve F is closed, we take the positive direction to be that corresponding to motion of the point ~E F so that the domain D bounded by the curve F will stay always to the left. Moreover, for the point ~a df df = ~b we taken an arbitrary point on the curve F, not a corner point, and we put [0 = ~a, ~k+l = ~b' The domain S for the variation of the arc-abscissa s for the case in which ~a ~ ~b is an interval in, b]; if ~a = ~b, we take the set S to be the whole real axis and the function ~(s) to be periodic with period 2l. For arbitrary e, dES we denoted by Fie, d] the arc of the curve F having endpoints at ~(c) and ~(d), ioeo P[c, dl = {~ : ; = ~ (s), s E [c, dl}. (2)
ABSTRACT For every n, we compute the Lebesgue constant of Rogosinski kernel with any preassigned ... more ABSTRACT For every n, we compute the Lebesgue constant of Rogosinski kernel with any preassigned accuracy.
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