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
This paper deals with approximation of smooth convex functions f on an interval by convex algebra... more This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of f ∈ ∆ (2) , the set of convex functions, from W r , the space of functions on [−1, 1] for which f (r−1) is absolutely continuous and f (r) ∞ := ess sup x∈[−1,1] |f (r) (x)| < ∞: For any f ∈ W r ∩ ∆ (2) , r ∈ N, there exists a number N = N(f, r), such that for every n ≥ N, there is an algebraic polynomial of degree ≤ n which is in ∆ (2) and such that
It is not surprising that one should expect that the degree of constrained (shape preserving) app... more It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-q-monotone approximation with q ≥ 3 is completely different from comonotone and coconvex cases. Additionally, we show that, for each function f from ∆ (1) , the set of all monotone functions on [−1, 1], and every α > 0, we have lim sup n→∞ inf Pn∈Pn∩∆ (1) n α (f − Pn) ϕ α ≤ c(α) lim sup n→∞ inf Pn∈Pn n α (f − Pn) ϕ α where Pn denotes the set of algebraic polynomials of degree < n, ϕ(x) := √ 1 − x 2 , and c = c(α) depends only on α.
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.$$ c\left(n,k\right)={\left(1+k\frac{\sqrt{1+{n}^2}-1}{n}\right)}^2-k. $$
The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights (1 − x) α... more The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights (1 − x) α (1 + x) β for functions in the Jacobi weighted L p [−1, 1], 0 < p ≤ ∞, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted L p spaces. If 1 ≤ p ≤ ∞, then these moduli are equivalent to certain weighted K-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these p), while for 0 < p < 1 they are equivalent to certain "Realization functionals".
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˛> 0 and a continuous function f; which is ... more In Part I of the paper, we have proved that, for every˛> 0 and a continuous function f; which is either convex .s D 0/ or changes convexity at a finite collection Y s D fy i g s iD1 of points y i 2. 1; 1/; supfn˛E .2/ n .f; Y s /W n N g Ä c.˛; s/ supfn˛E n .f /W n 1g; 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.˛; s/ is a constant depending only on˛and s: Moreover, it has been shown that N may be chosen to be 1 for s D 0 or s D 1;˛¤ 4; and that it must depend on Y s and˛for s D 1; D 4 or s 2: In Part II of the paper, we show that a more general inequality supfn˛E .2/ n .f; Y s /W n N g Ä c.˛; N ; s/ supfn˛E n .f /W n N g;
We introduce new moduli of smoothness for functions f ∈ L p [−1, 1]∩C r−1 (−1, 1), 1 ≤ p ≤ ∞, r ≥... more We introduce new moduli of smoothness for functions f ∈ L p [−1, 1]∩C r−1 (−1, 1), 1 ≤ p ≤ ∞, r ≥ 1, that have an (r − 1)st locally absolutely continuous derivative in (−1, 1), and such that ϕ r f (r) is in L p [−1, 1], where ϕ(x) = (1 − x 2) 1/2. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in L p [−1, 1] (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials. * AMS classification: 41A10, 41A17, 41A25. Keywords and phrases: Approximation by polynomials in the L p-norm, Degree of approximation, Jackson-type estimates, moduli of smoothness.
We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of conti... more We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of continuous and r-times continuously di erentiable functions which change their monotonicity nitely many times in a nite interval, say ?1;1], with special attention to the constants involved in the estimates. We describe four possibilities ranging from the existence of estimates involving absolute constants and which are valid for polynomials of all degrees except the very few rst ones, through estimates where either the constants or the degrees of the polynomials depend on the location of the points of change of monotonicity, then through estimates where either the constants or the degrees of the polynomials depend on the speci c function, and nally cases where there are no Jackson type estimates possible.
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&quot; We assume the curve F to be given in the parametric form: r= ... more ABSTRACT points ~a and ~b&quot; 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&#39; 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)
UDC 517.53 Dzydyk's studies in the area of mathematical analysis and its applications are well kn... more UDC 517.53 Dzydyk's studies in the area of mathematical analysis and its applications are well known to a broad group of specialists. In the present article, we will attempt to describe basically those of his results that belong to the constructive theory of functions of a complex variable. We are fully justified in considering Dzydyk's principal papers in this area as fundamental; in the 30 years following publication of his first articles [5-7] of the approximation of continuous functions of a complex variable in closed domains with angles, the concepts presented in these articles defined the direction of the research of the mathematical school he created as well as that of many mathematicians in the Soviet Union and abroad on the solution of the intricate and difficult problem of direct and inverse theorems in the constructive theory of functions of a complex variable. Dzydyk obtained a number of profound and conclusive results in other areas of complex analysis, for example, analytic and harmonic transformations, limiting values of Cauchy-type integrals, Dirichlet series, Pade approximation, and the problem of moments, and also strengthened several classical results. One monograph that has achieved appreciable renown is [I], a paper which exerted a major influence on the development of the scientific interest of specialists on the theory of approximation of functions as well as novice mathematical researchers.
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
This paper deals with approximation of smooth convex functions f on an interval by convex algebra... more This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of f ∈ ∆ (2) , the set of convex functions, from W r , the space of functions on [−1, 1] for which f (r−1) is absolutely continuous and f (r) ∞ := ess sup x∈[−1,1] |f (r) (x)| < ∞: For any f ∈ W r ∩ ∆ (2) , r ∈ N, there exists a number N = N(f, r), such that for every n ≥ N, there is an algebraic polynomial of degree ≤ n which is in ∆ (2) and such that
It is not surprising that one should expect that the degree of constrained (shape preserving) app... more It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-q-monotone approximation with q ≥ 3 is completely different from comonotone and coconvex cases. Additionally, we show that, for each function f from ∆ (1) , the set of all monotone functions on [−1, 1], and every α > 0, we have lim sup n→∞ inf Pn∈Pn∩∆ (1) n α (f − Pn) ϕ α ≤ c(α) lim sup n→∞ inf Pn∈Pn n α (f − Pn) ϕ α where Pn denotes the set of algebraic polynomials of degree < n, ϕ(x) := √ 1 − x 2 , and c = c(α) depends only on α.
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. $$
The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights (1 − x) α... more The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights (1 − x) α (1 + x) β for functions in the Jacobi weighted L p [−1, 1], 0 < p ≤ ∞, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted L p spaces. If 1 ≤ p ≤ ∞, then these moduli are equivalent to certain weighted K-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these p), while for 0 < p < 1 they are equivalent to certain "Realization functionals".
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˛> 0 and a continuous function f; which is ... more In Part I of the paper, we have proved that, for every˛> 0 and a continuous function f; which is either convex .s D 0/ or changes convexity at a finite collection Y s D fy i g s iD1 of points y i 2. 1; 1/; supfn˛E .2/ n .f; Y s /W n N g Ä c.˛; s/ supfn˛E n .f /W n 1g; 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.˛; s/ is a constant depending only on˛and s: Moreover, it has been shown that N may be chosen to be 1 for s D 0 or s D 1;˛¤ 4; and that it must depend on Y s and˛for s D 1; D 4 or s 2: In Part II of the paper, we show that a more general inequality supfn˛E .2/ n .f; Y s /W n N g Ä c.˛; N ; s/ supfn˛E n .f /W n N g;
We introduce new moduli of smoothness for functions f ∈ L p [−1, 1]∩C r−1 (−1, 1), 1 ≤ p ≤ ∞, r ≥... more We introduce new moduli of smoothness for functions f ∈ L p [−1, 1]∩C r−1 (−1, 1), 1 ≤ p ≤ ∞, r ≥ 1, that have an (r − 1)st locally absolutely continuous derivative in (−1, 1), and such that ϕ r f (r) is in L p [−1, 1], where ϕ(x) = (1 − x 2) 1/2. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in L p [−1, 1] (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials. * AMS classification: 41A10, 41A17, 41A25. Keywords and phrases: Approximation by polynomials in the L p-norm, Degree of approximation, Jackson-type estimates, moduli of smoothness.
We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of conti... more We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of continuous and r-times continuously di erentiable functions which change their monotonicity nitely many times in a nite interval, say ?1;1], with special attention to the constants involved in the estimates. We describe four possibilities ranging from the existence of estimates involving absolute constants and which are valid for polynomials of all degrees except the very few rst ones, through estimates where either the constants or the degrees of the polynomials depend on the location of the points of change of monotonicity, then through estimates where either the constants or the degrees of the polynomials depend on the speci c function, and nally cases where there are no Jackson type estimates possible.
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&quot; We assume the curve F to be given in the parametric form: r= ... more ABSTRACT points ~a and ~b&quot; 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&#39; 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)
UDC 517.53 Dzydyk's studies in the area of mathematical analysis and its applications are well kn... more UDC 517.53 Dzydyk's studies in the area of mathematical analysis and its applications are well known to a broad group of specialists. In the present article, we will attempt to describe basically those of his results that belong to the constructive theory of functions of a complex variable. We are fully justified in considering Dzydyk's principal papers in this area as fundamental; in the 30 years following publication of his first articles [5-7] of the approximation of continuous functions of a complex variable in closed domains with angles, the concepts presented in these articles defined the direction of the research of the mathematical school he created as well as that of many mathematicians in the Soviet Union and abroad on the solution of the intricate and difficult problem of direct and inverse theorems in the constructive theory of functions of a complex variable. Dzydyk obtained a number of profound and conclusive results in other areas of complex analysis, for example, analytic and harmonic transformations, limiting values of Cauchy-type integrals, Dirichlet series, Pade approximation, and the problem of moments, and also strengthened several classical results. One monograph that has achieved appreciable renown is [I], a paper which exerted a major influence on the development of the scientific interest of specialists on the theory of approximation of functions as well as novice mathematical researchers.
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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