Exact $L_2$ Bernstein-Markov inequalities for generalized weights

Heping Wang; Jiansong Li; Jiaxin Geng; Yun Ling

REVIEW 2 major objections 4 minor 17 references

Exact $L_2$ Bernstein-Markov inequalities for generalized weights

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper determines the exact L2 Bernstein–Markov constants for generalized Hermite and Gegenbauer weights, for both the ordinary derivative and the Dunkl operator, and identifies the polynomials that attain them.

desk verdict Exact L2 Bernstein-Markov constants are correct, but the extremal-polynomial uniqueness claims fail at eigenvalue coincidences; the paper is worth refereeing after a revision. read the letter →

arxiv 2411.16359 v1 pith:YGCJLI7O submitted 2024-11-25 math.CA

Jiansong Li , Jiaxin Geng , Yun Ling , Heping Wang This is my paper · ORCID

classification math.CA MSC 33C4541A1741A4442C05

keywords L2Bernstein-MarkovinequalitiesgeneralizedHermiteweightGegenbauerDunkloperatorextremalpolynomialsexactconstantsorthogonal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper determines, for every degree $n$, the sharp constant in the weighted-$L_2$ comparison between a polynomial and its derivative (or Dunkl derivative), for two families of even weights: the generalized Hermite weight $|x|^{2\lambda}e^{-x^2}$ on the real line and the generalized Gegenbauer weight $|x|^{2\lambda}(1-x^2)^{\mu-1/2}$ on $[-1,1]$. For the ordinary derivative the constant is $\sqrt{2n}$ when $n$ is even (the same as for the classical Hermite weight), and for odd $n$ it is the largest positive root of an explicit determinant; in the Gegenbauer case it is the maximum of such a root and $\sqrt{n(n+2\lambda+2\mu)}$ (or a shifted version for odd $n$). For the Dunkl operator the constants collapse to simple closed forms: $\sqrt{2n}$, $\sqrt{2(n+2\lambda-1)}$, or $\sqrt{2(n+2\lambda)}$ for Hermite weights depending on parity and on $\lambda$, and $\sqrt{n(n+2\lambda+2\mu)}$ with stated corrections for Gegenbauer weights. In every case the extremal polynomials are identified as generalized Hermite or Gegenbauer polynomials, or odd polynomials from a specified linear system, so the inequalities are sharp.

What carries the argument

The argument rests on a duality lemma (Lemmas 1 and 2) that equates the Bernstein–Markov factor with the largest $M>0$ for which a differential equation has a nontrivial polynomial solution: for the ordinary derivative, the integral equation system $\int_I \{A p'' + C p' + (2\lambda/x) p' + M^2 p\} q\, W_\lambda = 0$ for all $q\in\mathcal{P}_n$, and for the Dunkl operator the differential equation $A D_\lambda^2 p + B D_\lambda p + M^2 p = 0$, with coefficients $A,B,C$ from Table 1. Propositions 1–4 classify all polynomial solutions of these singular equations, using coefficient recurrences (2.4) and (2.9) from [4], as scalar multiples of the generalized Gegenbauer polynomials $C_s^{(\mu,\lambda)}$ or generalized Hermite polynomials $H_s^\lambda$, with eigenvalues given by (2.6) and (2.11). The parity decomposition of the extremal polynomial then forces either an even solution, which is a generalized orthogonal polynomial, or an odd one; in the odd case the constant appears as the largest positive root of an explicit determinant $F$ or $G$ built from the even moments of the weight.

What would settle it

For a fixed $\lambda>0$ and an odd degree $n$ (for instance $\lambda=1$, $n=3$), directly compute the largest eigenvalue of the matrix with entries $\int_{\mathbb{R}} p_i' p_j' w_\lambda\,\mathrm{d}x$ relative to $\int_{\mathbb{R}} p_i p_j w_\lambda\,\mathrm{d}x$ over a basis of $\mathcal{P}_3$; the claimed constant is $\sqrt{\nu_2}$ from Example 1, and any ratio exceeding that value, or a largest eigenvalue differing from $\nu_2$, would refute Theorem 1.

Watch

Extended reading notes

Core claim

The paper's central claim is that the extremal problem $M_n^2(L_2(W_\lambda), D) = \sup_{p\in\mathcal{P}_n} \|D p\|_{L_2(W_\lambda)}^2 / \|p\|_{L_2(W_\lambda)}^2$ has exact, explicitly computable values. For the generalized Hermite weight $w_\lambda(x)=|x|^{2\lambda}e^{-x^2}$ on $\mathbb{R}$ with $D=\mathrm{d}/\mathrm{d}x$, $M_n=\sqrt{2n}$ for even $n$, while for odd $n$ $M_n=\sqrt{\nu_{(n+1)/2}}$ where $\nu_m$ is the largest positive root of the determinant $F_m(t)=\det\{(2j+1)(2j+2\lambda)d_{2i+2j}+(t-4j-2)d_{2i+2j+2}\}_{i,j=0}^{m-1}$ with moments $d_{2s}=\Gamma(s+\lambda+1/2)$. For the generalized Gegenbauer weight $w_{\lambda,\mu}(x)=|x|^{2\lambda}(1-x^2)^{\mu-1/2}$ on $[-1,1]$ with $D=(1-x^2)^{1/2}\mathrm{d}/\mathrm{d}x$, the constant is the maximum of $\sqrt{\nu}$ and $\sqrt{n(n+2\lambda+2\mu)}$ for even $n$ (and of $\sqrt{\nu}$ with $\sqrt{(n-1)(n+2\lambda+2\mu-1)}$ for odd $n$), where $\nu$ is the largest positive root of the analogous determinant $G_m$ built from the $\beta$ moments $c_{2s}=\Gamma(s+\lambda+1/2)\Gamma(\mu+1/2)/\Gamma(\lambda+\mu+s+1)$. For the Dunkl operator $D_\lambda$, the constants are exactly $\sqrt{2n}$, $\sqrt{2(n+2\lambda-1)}$, or $\sqrt{2(n+2\lambda)}$ in the Hermite case depending on parity and on whether $\lambda\le 1/2$, and for the Gegenbauer case $\sqrt{n(n+2\lambda+2\mu)+4\lambda\mu}$ for odd $n$, plus the stated $\sqrt{n(n+2\lambda+2\mu)}$ or $\sqrt{n(n+2\lambda+2\mu)+2(n_0-n)}$ alternatives for even $n$ when $(2\lambda-1)(2\mu-1)>4$, with $n_0=(\lambda-1/2)(2\mu-1)$. Each constant is attained by the stated generalized orthogonal polynomial (or, in the odd ordinary-derivative cases, by an odd polynomial solving the displayed linear system), so all displayed inequalities are sharp.

Load-bearing premise

The full classification of polynomial solutions of the singular differential equations in Propositions 1–4 is taken from the coefficient recurrences (2.4) and (2.9) of reference [4]; if that classification missed any mixed-parity or additional polynomial solutions, the claimed maxima could be too low.

Editorial extensions

If this is right

The classical constants are recovered as special cases: $\lambda=0$ in Theorem 3 gives Schmidt's $\sqrt{2n}$ for the Hermite weight, and $\lambda=0$ in Theorem 4 gives the Gegenbauer constant $\sqrt{n(n+2\mu)}$.
For the ordinary derivative and even $n$, the generalized Hermite constant is $\sqrt{2n}$, independent of the singularity parameter $\lambda$.
For odd $n$, the exact constant is the largest positive root of a determinant of size $(n+1)/2$ (ordinary derivative), so it is computable in finitely many algebraic operations but not a simple closed formula in general.
In the Dunkl/Gegenbauer case, the extremal polynomial switches from $C_n^{(\mu,\lambda)}$ to $C_{n-1}^{(\mu,\lambda)}$ when $(2\lambda-1)(2\mu-1)>4$ and $n$ is below the threshold $n_0=(\lambda-1/2)(2\mu-1)$.
The quadratic inequalities of Theorem 5, with their equality cases, characterize the generalized Gegenbauer and Hermite polynomials as the unique extremizers, extending the $\lambda=0$ results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The parity-splitting determinant construction suggests a general recipe for any even weight: restrict to the odd subspace, and the largest eigenvalue of the resulting moment matrix gives the Bernstein–Markov constant whenever the even subspace is governed by a classified orthogonal family.
For the generalized Hermite weight, one could test numerically whether the odd-$n$ roots $\nu_{(n+1)/2}$ approach $2n$ as $n\to\infty$; the paper does not address the size of the parity gap.
The same duality lemma is specific to $L_2$; for $L_q$ with $q\ne2$ no such eigenvalue/determinant characterization is known, so the exact-constant phenomenon described here is likely special to the Hilbert-space setting.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the exact constants are right, but the extremal-polynomial claims are overreaching at a few degenerate parameter values. The paper deserves a serious referee, and a revision that fixes the uniqueness statements.

What's new: exact L2 Bernstein-Markov factors for generalized Hermite and Gegenbauer weights, both for the ordinary derivative and for Dunkl operators. For λ>0 and ordinary derivative, only two-sided bounds were known (Draux–Kaliaguine), so Theorems 1–2 are a real step. Theorems 3–4 for Dunkl operators are new and reduce to known Schmidt and Guessab–Milovanovic results at λ=0. The proofs are clean: duality lemmas convert the extremal ratio into an eigenproblem, and coefficient recurrences from Ben Cheikh–Gaied identify the eigenvalues. I checked the algebra in the propositions; the eigenvalue computation is sound, and the determinant formulas for the odd cases look right.

The problem is the uniqueness clauses. Propositions 1 and 3 claim every polynomial solution is a scalar multiple of one generalized Gegenbauer or Hermite polynomial. That is false when two eigenvalues coincide across parities. Example: for the Dunkl Gegenbauer equation with λ=1, μ=5/2, both C_1 and C_2 have λ^2=18, so any linear combination solves. The same happens in the Dunkl Hermite case at λ=1/2, where H_{n-1} and H_n are both extremal for even n. This makes the extremal-polynomial statements in Theorem 3 (Table 3), Theorem 4(ii)–(iii) and its Table 4, and the “if and only if” in Theorem 5 overclaimed at those boundaries. The constants are not endangered: taking the max over eigenvalues still gives the right M_n, so the main theorems survive. But the characterization of extremizers is wrong as stated, and some reader will trip on it.

One thing the reader's report worried about—the reliance on the external classification of polynomial solutions—doesn't bother me. Even/odd subspaces decouple, and the first-order recurrences force the eigenvalue list; the stress-test note says the same and I agree.

Bottom line: worth a serious referee. Ask the authors to rework Propositions 1 and 3 to describe the solution space at eigenvalue coincidences, and to qualify the equality cases in Theorems 3–5 accordingly. The main constants are a solid contribution.

Referee Report

2 major / 4 minor

Summary. The paper derives exact L2 Bernstein-Markov factors for generalized Hermite weights |x|^{2λ}e^{-x^2} and generalized Gegenbauer weights |x|^{2λ}(1-x^2)^{μ-1/2}, for both the ordinary derivative and the Dunkl operator. The main results (Theorems 1-4) express the sharp constant as the square root of the largest eigenvalue of an explicitly given differential or differential-difference operator, with the extremal polynomials identified as generalized Hermite or Gegenbauer polynomials. The proofs rest on two duality lemmas that reformulate the supremum as an eigenvalue problem, followed by coefficient-recurrence arguments and determinant formulas for the odd-degree cases. The paper also contains a characterization-type inequality for the generalized orthogonal polynomials (Theorem 5) and several worked examples.

Significance. If correct, the paper closes a gap in the literature: for λ>0 the ordinary-derivative Hermite case was previously known only via bounds, and the Dunkl-operator cases appear to be new. The method is attractive and self-contained in its main steps: the duality lemmas are clean, the coefficients of the generalized polynomials are derived from explicit recurrences, and no parameters are fitted. The exact constants themselves survive scrutiny. However, the paper's extremal-polynomial classification and its equality characterization are overstated, because the relevant eigenvalue problems have nontrivial degeneracies at certain parameter values. The central numerical constants are not endangered, but the statements describing all extremal polynomials must be weakened and the degeneracy cases recorded.

major comments (2)

[Section 2, Proposition 1] The uniqueness claim that a nontrivial polynomial solution p of (2.7) must be a scalar multiple of a single C_s is false when the eigenvalues in (2.6) coincide for indices of different parity. For example, with λ=1 and μ=5/2, one has λ_1^2=λ_2^2=18, so p = a C_1^{(5/2,1)} + b C_2^{(5/2,1)} solves (2.7) with M^2=18 but is not proportional to C_s for any single s. The proof's deduction 'a_{s-1}=a_{s-3}=...=0' on page 8 is not justified because the recurrence coefficient can vanish exactly when M^2 equals the opposite-parity eigenvalue. The same defect occurs in Proposition 3 for the Hermite equation (2.12); for λ=1/2, λ_1^2=λ_2^2=4, so H_1 and H_2 are both solutions for the same M^2.
[Theorems 3-4] The degeneracy propagates to the extremal-polynomial statements. In Theorem 3 and Table 3, for even n and λ=1/2, both H_{n-1} and H_n attain the value M_n^2=2n, so the full extremal set includes their linear combinations, not only cH_n as stated. Likewise, in Theorem 4(ii) at the boundary (2λ-1)(2μ-1)=4 with n=2, λ_2^2=λ_1^2, so additional extremal polynomials exist beyond cC_2. The equality characterization in Theorem 5 should be 'p belongs to the span of {C_s : λ_s^2=λ_n^2}' rather than 'p=cC_n'. None of this lowers the exact constants in Theorems 1-4: for any nontrivial solution, the highest-degree parity component is itself a solution, and the leading-coefficient argument still forces M^2 to be one of the listed eigenvalues. The constants are therefore sound, but the uniqueness and 'if and only if' claims require correction.

minor comments (4)

[Proposition 4] The statement says 'Let λ>0 and μ>-1/2' but the equation (2.13) contains no μ; the condition on μ is extraneous and should be removed.
[References] Reference [16] misspells the author's name as 'Schimidt'; it should be 'Schmidt'.
[Table 2] The column header 'λ(λ = -μ)' is potentially confusing because the table also allows positive μ; a short note describing the parameter ranges used in the computations would improve readability.
[Theorem 1 and Theorem 2] The determinant definitions of F_{m+1}(t) and G_{m+1}(t) are terse; stating the matrix size and the range of indices explicitly in the theorem statements would help readers verify the Cramer-rule step.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the Bernstein-Markov constants are obtained from explicit eigenvalue problems, with only external, non-self coefficient recurrences cited.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity review. Lemma 1 and Lemma 2 prove duality relations that identify the extremal Bernstein-Markov factor with the largest M for which an explicitly displayed differential or integro-differential equation has a nontrivial polynomial solution. Propositions 1-4 then determine all polynomial solutions by coefficient recurrences, and the theorems take maxima over the resulting explicit eigenvalues. No parameter is fitted to the target data, no prediction is post hoc, and no result is assumed equivalent to the conclusion. The determinant formulas in Theorems 1 and 2 (e.g., F_m(t), G_m(t)) are explicit algebraic characterizations of the eigenvalue problem: their matrix entries are built from known moments and recurrence coefficients, not from the unknown value of M_n. Thus computing the largest positive root of such a determinant is a genuine reduction, not a self-definitional restatement. The classification of polynomial solutions relies on the coefficient recurrences (2.4) and (2.9) taken from reference [4], but that reference is external to the present authors and the recurrences are standard characterizations of generalized Gegenbauer and Hermite polynomials; moreover, the paper re-derives the solution classification from those recurrences in the proofs of Propositions 1-4. There is no self-citation chain, no uniqueness theorem imported from the authors' own prior work, and no ansatz smuggled in under the guise of a citation. Any concern about degenerate parameter choices producing non-uniqueness at coincident eigenvalues (e.g., lambda_s^2 = lambda_{s'}^2) is a correctness issue about the uniqueness clause in Propositions 1-4 and Remark 3, not a circularity: it does not make the claimed extremal constants equal to the inputs by construction. Overall, the paper's exact-value claims have independent mathematical content and are not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on known properties of generalized Gegenbauer and Hermite polynomials and on standard integration-by-parts identities; no free parameters are fitted to data and no new entities are postulated.

assumptions (4)

domain assumption The generalized Gegenbauer polynomials C_n^{(μ,λ)} and generalized Hermite polynomials H_n^λ form orthogonal bases of P_n for their weights and satisfy the differential equations (2.5) and (2.10) with eigenvalues (2.6) and (2.11).
Invoked in Propositions 1-4 to identify all polynomial eigenfunctions; cited from reference [4] (Ben Cheikh-Gaied) and [7].
domain assumption The weight functions satisfy the Pearson-type equation d/dx(A(x)Wλ(x)) = B(x)Wλ(x) with B(x)=B'(0)x, and A Wλ vanishes at the endpoints and at 0 for λ>0.
Needed for the integration-by-parts identities (2.2), (2.3), (3.4), (3.6), (4.6), and (4.11). Verified for the listed weights in Table 1.
standard math The Rayleigh quotient sup over P_n is attained.
Finite-dimensional projective space is compact; used to choose an extremal polynomial p* in Lemmas 1-2.
standard math A polynomial of degree at most n that is orthogonal to all of P_n with respect to a positive weight is identically zero.
Used in Lemmas 1-2 to pass from integral orthogonality to the pointwise differential equation satisfied by extremal polynomials.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Exact $L_2$ Bernstein-Markov inequalities for generalized weights." pith.science (2026). https://pith.science/paper/YGCJLI7O

@misc{pith2026241116359,
  author       = {Pith},
  title        = {Pith review of: Exact $L_2$ Bernstein-Markov inequalities for generalized weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGCJLI7O}},
  note         = {Machine review of arXiv:2411.16359}
}

abstract

In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n^2(L_2(W_\lambda),{\rm D}):=\sup_{0\neq p\in\mathcal{P}_n}\frac{\int_I\left|{\rm D} p(x)\right|^2W_\lambda(x){\rm d}x}{\int_I| p(x)|^2W_\lambda(x){\rm d}x},\ \lambda>0,$$ where $\mathcal{P}_n$ denotes the set of all algebraic polynomials of degree at most $n$, ${\rm D}$ is the differential operator given by $${\rm D}=\Bigg\{\begin{aligned}&\frac {\rm d}{{\rm d}x}\ {\rm or}\ \mathcal{D}_\lambda, &&{\rm if}\ W_\lambda(x)=|x|^{2\lambda}e^{-x^2}\ {\rm and}\ I=\mathbb R, \\&(1-x^2)^{\frac12}\,\frac {\rm d}{{\rm d}x}\ {\rm or}\ (1-x^2)^{\frac12}\,\mathcal{D}_\lambda, &&{\rm if}\ W_\lambda(x):=|x|^{2\lambda}(1-x^2)^{\mu-\frac 12},\mu>-\frac12\ {\rm and}\ I=[-1,1],\end{aligned} $$ and $\mathcal{D}_\lambda$ is the univariate Dunkl operator, i.e., $\mathcal{D}_\lambda f(x)=f'(x)+\lambda{(f(x)-f(-x))}/{x}$. Furthermore, the corresponding extremal polynomials are also obtained.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Arestov, On integral inequalities for trigonometric polynomia ls and their derivatives, Izv

V.V. Arestov, On integral inequalities for trigonometric polynomia ls and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat., 1 (45) (1981), 3-22

work page 1981
[2]

Agarwal, G

R.P. Agarwal, G. V. Milovanovi´ c, A characterization of the class ical orthogonal polynomials, in: Progress in Approximation Theory (P. Nevai, A. Pinkus, eds.), Academic Press , New York, 1991, pp. 1-4

work page 1991
[3]

Agarwal, G.V

R.P. Agarwal, G.V. Milovanovic, Extremal problems, inequalities, an d classical orthogonal polynomials. Orthogonal systems and applications, Appl. Math. Comput., 128 (2 002), 151-166

work page
[4]

Ben Cheikh, M

Y. Ben Cheikh, M. Gaied, Characterization of the Dunkl-classical symmetric orthogonal polynomials, Appl. Math. Comput., 187 (2007), 105-114

work page 2007
[5]

S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions con tinues par les polynomes de degr´ e donn´ e, M´ emoires publi´ es par la Classe des Sciences del l’Acad´ emie de Belgique, (1912), 4. EXACT L2 BERNSTEIN-MARKOV INEQUALITIES 23

work page 1912
[6]

Draux, V

A. Draux, V. Kaliaguine, Markov-Bernstein inequalities for gener alized Hermite weight, East J. Approx., 12 (1) (2006), 1-23

work page 2006
[7]

Dunkl, Y

C. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Se cond Edition, Encyclopedia of Mathe- matics and its Applications, vol. 155, Cambridge Univ. Press, 2014

work page 2014
[8]

Dzyadyk, Introduction to the Theory of Uniform Approxima tion of Functions by Polynomials, Nauka, Moscow, 1977 (in Russian)

V.K. Dzyadyk, Introduction to the Theory of Uniform Approxima tion of Functions by Polynomials, Nauka, Moscow, 1977 (in Russian)

work page 1977

Show all 17 references

[9]

Gardner, N.K

R.B. Gardner, N.K. Govil, G.V. Milovanovic, Extremal Problems and I nequalities of Markov-Bernstein Type for Algebraic Polynomials, Elsevier/Academic Press, London, 2 022
[10]

Guessab, G.V

A. Guessab, G.V. Milovanovic, Weighted L2-analogous of Bernstein’s inequality and classical orthogonal polynomials, J. Math. Anal. Appl., 182 (1994), 244-249

1994
[11]

Hille, G, Szeg¨ o, I.D

E. Hille, G, Szeg¨ o, I.D. Tamarkin, On some generalizations of a th eorem of A. Markoﬀ, Duke Math. J., 3 (4) (1937), 729-739

1937
[12]

Ivanov, Some extremal problems for polynomials and invers e inequalities in approximation theory, Tr

V.I. Ivanov, Some extremal problems for polynomials and invers e inequalities in approximation theory, Tr. Mat. Inst. Steklova 145 (1979), 79-110 (in Russian)

1979
[13]

Lorentz, Approximation of Functions, Holt, Reinhart and W inston, New York, 1966

G.G. Lorentz, Approximation of Functions, Holt, Reinhart and W inston, New York, 1966

1966
[14]

Markov, Sur une question posee par Mendeleieﬀ , Bull

A.A. Markov, Sur une question posee par Mendeleieﬀ , Bull. Acad. Sci. St. Petersburg 62 (1889), 1-24
[15]

Meinardus, Approximation von Funktionen und ihre numerische Behandlu ng, SpringerVerlag, Berlin, Gottingen-Heidelberg-New York, 1964

G. Meinardus, Approximation von Funktionen und ihre numerische Behandlu ng, SpringerVerlag, Berlin, Gottingen-Heidelberg-New York, 1964

1964
[16]

Preussischen Akad

Schimidt, Die asymptotiche bestimmung des maximums des integrals, Si tz.ber. Preussischen Akad. Wiss. 287 (1932)

1932
[17]

Tur´ an, Remark on a theorem of Erhard Schmidt, Mathemat ica (Cluj) 2 (1960), 373-378

P. Tur´ an, Remark on a theorem of Erhard Schmidt, Mathemat ica (Cluj) 2 (1960), 373-378. School of Mathematical Sciences, Capital Normal Universit y, Beijing 100048, China Email address : cnuljs2023@163.com School of Mathematical Sciences, Capital Normal Universit y, Beijing ...

1960

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Arestov, On integral inequalities for trigonometric polynomia ls and their derivatives, Izv

V.V. Arestov, On integral inequalities for trigonometric polynomia ls and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat., 1 (45) (1981), 3-22

work page 1981

[2] [2]

Agarwal, G

R.P. Agarwal, G. V. Milovanovi´ c, A characterization of the class ical orthogonal polynomials, in: Progress in Approximation Theory (P. Nevai, A. Pinkus, eds.), Academic Press , New York, 1991, pp. 1-4

work page 1991

[3] [3]

Agarwal, G.V

R.P. Agarwal, G.V. Milovanovic, Extremal problems, inequalities, an d classical orthogonal polynomials. Orthogonal systems and applications, Appl. Math. Comput., 128 (2 002), 151-166

work page

[4] [4]

Ben Cheikh, M

Y. Ben Cheikh, M. Gaied, Characterization of the Dunkl-classical symmetric orthogonal polynomials, Appl. Math. Comput., 187 (2007), 105-114

work page 2007

[5] [5]

S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions con tinues par les polynomes de degr´ e donn´ e, M´ emoires publi´ es par la Classe des Sciences del l’Acad´ emie de Belgique, (1912), 4. EXACT L2 BERNSTEIN-MARKOV INEQUALITIES 23

work page 1912

[6] [6]

Draux, V

A. Draux, V. Kaliaguine, Markov-Bernstein inequalities for gener alized Hermite weight, East J. Approx., 12 (1) (2006), 1-23

work page 2006

[7] [7]

Dunkl, Y

C. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Se cond Edition, Encyclopedia of Mathe- matics and its Applications, vol. 155, Cambridge Univ. Press, 2014

work page 2014

[8] [8]

Dzyadyk, Introduction to the Theory of Uniform Approxima tion of Functions by Polynomials, Nauka, Moscow, 1977 (in Russian)

V.K. Dzyadyk, Introduction to the Theory of Uniform Approxima tion of Functions by Polynomials, Nauka, Moscow, 1977 (in Russian)

work page 1977