Proposed problems

Tiberiu Popoviciu
divided differences, paper, works

Abstract

Authors

R. Askey, R. DeVore, G. Freud, J. Musielak, J. Peetre

Tiberiu Popoviciu

Tiberiu Popoviciu

Keywords

?

Paper coordinates

Proposed problems, Contributed by R. Askey, R. DeVore, G. Freud, J. Musielak, J. Peetre and T. Popoviciu, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), pp. 533–538, Akadémiai Kiadó, Budapest, 1972.

PDF

1972 g -Popoviciu- Proceedings – Proposed problems

About this paper

Journal

Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969)

Publisher Name

published by Akadémiai Kiadó, Budapest

DOI

 

Print ISSN

 

Online ISSN

 

[MR 52#6245]

[/vc_column]

[1] G. Kowalewski, Interpolation und genäherte Quadrature, 1932.
[2] R. Mises, Über allgemeine Quadraturformeln, J. f. die reine u. angew. Math. 174 (1936) 56-67.
[3] T. Popoviciu, Notes sur les fonctions convexes d’ordre supérieur (IX). Bull. Math. Soc. Roumaine des sciences, 43 (1942) 84-141.
[4] T. Popoviciu, Sur le reste dans certaines formules linéaires d’approximation de l’analyse, Mathematica 1 (24) (1959) 95-142.
[5] T. Popoviciu, La simplicité du reste dans certaines formules de quadrature, Mathematica, 6 (29) (1964) 157-184.
[6]E. Rémèz, Sur certaines classes de fonctionnelles linéaires dans les espaces  et sur les termes complémentaires des formules d’analyse approximative I, Rec. Trav. Inst. Math. Acad. Sci. URSS Ukraine 3 (1940) 21-62
[7] E. Rémèz, Sur certaines classes de fonctionnelles linéaires dans les espaces  et sur les termes complémentaires des formules d’analyse approximative II. ibid., 4. (1940) 47-82.
[9] ” : ibid., 12, 81-92 (1936)
[10] ” : Cucrarile Ses. Gen. Stuntifice a Acad. R. P.R., 1950, p.183-185.
[11] ” : Mathematica, 1 (24), 95-142 (1960)
[12] Radon, J.: Monatshefte f.Math.u.Phys., 42, 389~396 (1935)[13] Ramez, E. Ya.: Zbirnik Praci Institutu Matem. Akad. Nauk. URSR, 3, 21-62 (1939)
[14] Sard, A.: Duke Math. J. 15, 333-345 (1948)
[15] Wigert, S.: Arkiv f. Mat. Astr. och Fysik, Bd. 22B, n. 9, 1-4 (1932).

Paper (preprint) in HTML form

1972 g -Popoviciu- Proceedings - Proposed problems

PROPOSED PROBLEMS

R. Askey

  1. Let P n − 1 ( x ) P n − 1 ( x ) P_(n-1)(x)P_{n-1}(x)Pn−1(x) be a polynomial of degree n − 1 , x k , n n − 1 , x k , n n-1,x_(k,n)n-1, x_{k, n}n−1,xk,n the zeros of the polynomials orthogonal with respect to d α ( x ) d α ( x ) d alpha(x)d \alpha(x)dα(x), a positive measure on [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1][−1,1], and λ k λ k lambda_(k)\lambda_{k}λk the corresponding Christoffel numbers. When is
[ ∑ k = 1 n | P n − 1 ( x k , n ) | p λ k ] 1 p ≦ A [ ∫ − 1 1 | P n − 1 ( x ) | p d α ( x ) ] 1 p ∑ k = 1 n   P n − 1 x k , n p λ k 1 p ≦ A ∫ − 1 1   P n − 1 ( x ) p d α ( x ) 1 p [sum_(k=1)^(n)|P_(n-1)(x_(k,n))|^(p)lambda_(k)]^((1)/(p)) <= A[int_(-1)^(1)|P_(n-1)(x)|^(p)d alpha(x)]^((1)/(p))\left[\sum_{k=1}^{n}\left|P_{n-1}\left(x_{k, n}\right)\right|^{p} \lambda_{k}\right]^{\frac{1}{p}} \leqq A\left[\int_{-1}^{1}\left|P_{n-1}(x)\right|^{p} d \alpha(x)\right]^{\frac{1}{p}}[∑k=1n|Pn−1(xk,n)|pλk]1p≦A[∫−11|Pn−1(x)|pdα(x)]1p
where A A AAA is independent of n n nnn and the polynomial P n − 1 ( x ) P n − 1 ( x ) P_(n-1)(x)P_{n-1}(x)Pn−1(x). For p = ∞ p = ∞ p=oop=\inftyp=∞ and p = 2 p = 2 p=2p=2p=2 it holds for all positive measures with A = 1 A = 1 A=1A=1A=1. It holds for 1 ≦ p ≦ ∞ 1 ≦ p ≦ ∞ 1 <= p <= oo1 \leqq p \leqq \infty1≦p≦∞ if d α ( x ) = ( 1 − x ) α ( 1 + x ) β d x d α ( x ) = ( 1 − x ) α ( 1 + x ) β d x d alpha(x)=(1-x)^(alpha)(1+x)^(beta)dxd \alpha(x)=(1-x)^{\alpha}(1+x)^{\beta} d xdα(x)=(1−x)α(1+x)βdx for some values of ( α , β ) ( α , β ) (alpha,beta)(\alpha, \beta)(α,β). For α = β = − 1 2 α = β = − 1 2 alpha=beta=-(1)/(2)\alpha=\beta=-\frac{1}{2}α=β=−12 this was proven by Marcinkiewicz and I can prove it for α = β > − 1 2 α = β > − 1 2 alpha=beta > -(1)/(2)\alpha=\beta>-\frac{1}{2}α=β>−12; α > β = − 1 2 ; α = k ; β = 0 ; α = 2 k + 1 , β = 1 ; α => , β = 3 , k α > β = − 1 2 ; α = k ; β = 0 ; α = 2 k + 1 , β = 1 ; α => , β = 3 , k alpha > beta=-(1)/(2);alpha=k;beta=0;alpha=2k+1,beta=1;alpha=>,beta=3,k\alpha>\beta=-\frac{1}{2} ; \alpha=k ; \beta=0 ; \alpha=2 k+1, \beta=1 ; \alpha=>, \beta=3, kα>β=−12;α=k;β=0;α=2k+1,β=1;α=>,β=3,k a positive integer.
2. Let A A AAA and B B BBB be two Banach spaces of functions on [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1][−1,1] with A ⊂ B A ⊂ B A sub BA \subset BA⊂B. Let P n ( x ) P n ( x ) P_(n)(x)P_{n}(x)Pn(x) be a polynomial of degree n n nnn. Assume that A A AAA contains all polynomials of degree n n nnn. Then
‖ Q n ‖ A ≦ a ( n , A , B ) ‖ P n ‖ B Q n A ≦ a ( n , A , B ) P n B ||Q_(n)||_(A) <= a(n,A,B)||P_(n)||_(B)\left\|Q_{n}\right\|_{A} \leqq a(n, A, B)\left\|P_{n}\right\|_{B}‖Qn‖A≦a(n,A,B)‖Pn‖B
Find the correct order of growth of a ( n , A , B ) a ( n , A , B ) a(n,A,B)a(n, A, B)a(n,A,B). My main interest is the case A = L q ( d α ) , B = L p ( d β ) A = L q ( d α ) , B = L p ( d β ) A=L^(q)(d alpha),B=L^(p)(d beta)A=L^{q}(d \alpha), B=L^{p}(d \beta)A=Lq(dα),B=Lp(dβ), but the problem is also of interest for Lipschitz spaces. (Markov's and Bernstein's inequalities are of this type.) A much harder question is to find the best constant. This is unknown even in the case A = L 4 , B = L 2 A = L 4 , B = L 2 A=L^(4),B=L^(2)A=L^{4}, B=L^{2}A=L4,B=L2 for trigonometric polynomials. See A. Garsin, E. Rodemich and H. Rumsey, "On some extremal positive definite functions", J. of Math. and Mech., 18 (1969), 805-834, p. 834, for some related results.
3. (Turán) Find a positive measure d α ( x ) d α ( x ) d alpha(x)d \alpha(x)dα(x) on [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1][−1,1] for which ∫ − 1 1 | f ( x ) − L n f ( x ) | p d α ( x ) → 0 ∫ − 1 1   f ( x ) − L n f ( x ) p d α ( x ) → 0 int_(-1)^(1)|f(x)-L_(n)^(f)(x)|^(p)d alpha(x)rarr0\int_{-1}^{1}\left|f(x)-L_{n}^{f}(x)\right|^{p} d \alpha(x) \rightarrow 0∫−11|f(x)−Lnf(x)|pdα(x)→0 fails for some continuous function for all p > 2 p > 2 p > 2p>2p>2. (Askey) I conjecture that the measures associated with Pollaczek
polynomials have this property. These measures vanish so rapidly at x = ± 1 x = ± 1 x=+-1x= \pm 1x=±1 that ∫ − 1 1 | log ⁡ w ( x ) | d x ( 1 − x 2 ) 1 2 ∫ − 1 1   | log ⁡ w ( x ) | d x 1 − x 2 1 2 int_(-1)^(1)(|log w(x)|dx)/((1-x^(2))^((1)/(2)))\int_{-1}^{1} \frac{|\log w(x)| d x}{\left(1-x^{2}\right)^{\frac{1}{2}}}∫−11|log⁡w(x)|dx(1−x2)12 diverges, d α ( x ) = w ( x ) d x d α ( x ) = w ( x ) d x d alpha(x)=w(x)dxd \alpha(x)=w(x) d xdα(x)=w(x)dx.
4. Prove that ∫ − 1 1 | L n f ( x , α , β ) − f ( x ) | p ( 1 − x ) γ ( 1 + x ) δ d x → 0 ∫ − 1 1   L n f ( x , α , β ) − f ( x ) p ( 1 − x ) γ ( 1 + x ) δ d x → 0 int_(-1)^(1)|L_(n)^(f)(x,alpha,beta)-f(x)|^(p)(1-x)^(gamma)(1+x)^(delta)dx rarr0\int_{-1}^{1}\left|L_{n}^{f}(x, \alpha, \beta)-f(x)\right|^{p}(1-x)^{\gamma}(1+x)^{\delta} d x \rightarrow 0∫−11|Lnf(x,α,β)−f(x)|p(1−x)γ(1+x)δdx→0 for all continuous functions if α ≥ β > − 1 α ≥ β > − 1 alpha >= beta > -1\alpha \geq \beta>-1α≥β>−1 and
(i) if α > − 1 2 α > − 1 2 quad alpha > -(1)/(2)\quad \alpha>-\frac{1}{2}α>−12 then p < min ( 4 ( γ + 1 ) / ( 2 α + 1 ) , 4 ( δ + 1 ) / ( 2 β + 1 ) ) p < min ( 4 ( γ + 1 ) / ( 2 α + 1 ) , 4 ( δ + 1 ) / ( 2 β + 1 ) ) p < min(4(gamma+1)//(2alpha+1),4(delta+1)//(2beta+1))p<\min (4(\gamma+1) /(2 \alpha+1), 4(\delta+1) /(2 \beta+1))p<min(4(γ+1)/(2α+1),4(δ+1)/(2β+1)),
(ii) if − 1 < α ≦ − 1 2 − 1 < α ≦ − 1 2 -1 < alpha <= -(1)/(2)-1<\alpha \leqq-\frac{1}{2}−1<α≦−12 then p < ∞ , γ ≧ α , δ ≧ β p < ∞ , γ ≧ α , δ ≧ β p < oo,gamma >= alpha,delta >= betap<\infty, \gamma \geqq \alpha, \delta \geqq \betap<∞,γ≧α,δ≧β.
This is known for γ = α , δ = β ; γ = δ = 0 , p = 2 ; γ = δ = 0 , p = 1 γ = α , δ = β ; γ = δ = 0 , p = 2 ; γ = δ = 0 , p = 1 gamma=alpha,delta=beta;gamma=delta=0,p=2;gamma=delta=0,p=1\gamma=\alpha, \delta=\beta ; \gamma=\delta=0, p=2 ; \gamma=\delta=0, p=1γ=α,δ=β;γ=δ=0,p=2;γ=δ=0,p=1. Condition (i) is best possible.
5. Let f ( x ) ∼ ∑ n = 0 ∞ a n P n ( α , β ) ( x ) , f ( x ) ≥ 0 f ( x ) ∼ ∑ n = 0 ∞   a n P n ( α , β ) ( x ) , f ( x ) ≥ 0 f(x)∼sum_(n=0)^(oo)a_(n)P_(n)^((alpha,beta))(x),f(x) >= 0f(x) \sim \sum_{n=0}^{\infty} a_{n} P_{n}^{(\alpha, \beta)}(x), f(x) \geq 0f(x)∼∑n=0∞anPn(α,β)(x),f(x)≥0 and α + β + 1 ≧ 0 α + β + 1 ≧ 0 alpha+beta+1 >= 0\alpha+\beta+1 \geqq 0α+β+1≧0. Prove that the ( C , α + β + 2 ) ( C , α + β + 2 ) (C,alpha+beta+2)(C, \alpha+\beta+2)(C,α+β+2) means are non-negative. For α = β = − 1 2 , α = β = 0 α = β = − 1 2 , α = β = 0 alpha=beta=-(1)/(2),alpha=beta=0\alpha=\beta=-\frac{1}{2}, \alpha=\beta=0α=β=−12,α=β=0, α = − β = 1 2 α = − β = 1 2 alpha=-beta=(1)/(2)\alpha=-\beta=\frac{1}{2}α=−β=12 this was shown by Fejér, and for α = β > − 1 2 α = β > − 1 2 alpha=beta > -(1)/(2)\alpha=\beta>-\frac{1}{2}α=β>−12 it was shown by Kogbetliantz.
6. Let d α ( x ) d α ( x ) d alpha(x)d \alpha(x)dα(x) be a positive measure on [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1][−1,1] and p n ( x ) p n ( x ) p_(n)(x)p_{n}(x)pn(x) the corresponding orthonormal polynomials. For which p p ppp do we have ∫ − 1 1 | f ( x ) − S n f ( x ) | p d α ( x ) → 0 ∫ − 1 1   f ( x ) − S n f ( x ) p d α ( x ) → 0 int_(-1)^(1)|f(x)-S_(n)^(f)(x)|^(p)d alpha(x)rarr0\int_{-1}^{1}\left|f(x)-S_{n}^{f}(x)\right|^{p} d \alpha(x) \rightarrow 0∫−11|f(x)−Snf(x)|pdα(x)→0, where S n f ( x ) S n f ( x ) S_(n)^(f)(x)S_{n}^{f}(x)Snf(x) is the n n nnn-th partial sum of the orthogonal series in p n ( x ) p n ( x ) p_(n)(x)p_{n}(x)pn(x). For
d α ( x ) = ( 1 − x ) α 1 ∏ i = 2 j − 1 | x − x i | α i ( 1 + x ) α j d x , α 1 , α j ≥ − 1 2 , α i ≥ 0 d α ( x ) = ( 1 − x ) α 1 ∏ i = 2 j − 1   x − x i α i ( 1 + x ) α j d x , α 1 , α j ≥ − 1 2 , α i ≥ 0 d alpha(x)=(1-x)^(alpha_(1))prod_(i=2)^(j-1)|x-x_(i)|^(alpha_(i))(1+x)^(alpha_(j))dx,alpha_(1),alpha_(j) >= -(1)/(2),alpha_(i) >= 0d \alpha(x)=(1-x)^{\alpha_{1}} \prod_{i=2}^{j-1}\left|x-x_{i}\right|^{\alpha_{i}}(1+x)^{\alpha_{j}} d x, \alpha_{1}, \alpha_{j} \geq-\frac{1}{2}, \alpha_{i} \geq 0dα(x)=(1−x)α1∏i=2j−1|x−xi|αi(1+x)αjdx,α1,αj≥−12,αi≥0
i = 2 , 3 , … , j − 1 , − 1 < x j − 1 < … < x 2 < 1 i = 2 , 3 , … , j − 1 , − 1 < x j − 1 < … < x 2 < 1 i=2,3,dots,j-1,-1 < x_(j-1) < dots < x_(2) < 1i=2,3, \ldots, j-1,-1<x_{j-1}<\ldots<x_{2}<1i=2,3,…,j−1,−1<xj−1<…<x2<1, I conjecture that the correct range is
4 ( 1 + α i ) / ( 2 α i + 3 ) < p < 4 ( 1 + α i ) / ( 2 α i + 1 ) , i = 1 , … , j 2 ( 1 + α i ) / ( α i + 2 ) < p < 2 ( 1 + α i ) / α i , i = 2 , … , j − 1 4 1 + α i / 2 α i + 3 < p < 4 1 + α i / 2 α i + 1 ,      i = 1 , … , j 2 1 + α i / α i + 2 < p < 2 1 + α i / α i ,      i = 2 , … , j − 1 {:[4(1+alpha_(i))//(2alpha_(i)+3) < p < 4(1+alpha_(i))//(2alpha_(i)+1)",",i=1","dots","j],[2(1+alpha_(i))//(alpha_(i)+2) < p < 2(1+alpha_(i))//alpha_(i)",",i=2","dots","j-1]:}\begin{array}{ll} 4\left(1+\alpha_{i}\right) /\left(2 \alpha_{i}+3\right)<p<4\left(1+\alpha_{i}\right) /\left(2 \alpha_{i}+1\right), & i=1, \ldots, j \\ 2\left(1+\alpha_{i}\right) /\left(\alpha_{i}+2\right)<p<2\left(1+\alpha_{i}\right) / \alpha_{i}, & i=2, \ldots, j-1 \end{array}4(1+αi)/(2αi+3)<p<4(1+αi)/(2αi+1),i=1,…,j2(1+αi)/(αi+2)<p<2(1+αi)/αi,i=2,…,j−1
Some case with d α ( x ) d α ( x ) d alpha(x)d \alpha(x)dα(x) a set of point masses should be worked out to see if this influences the range of p p ppp, or whether it is only the zeros of ( 1 − x 2 ) 1 2 w ( x ) d x 1 − x 2 1 2 w ( x ) d x (1-x^(2))^((1)/(2))w(x)dx\left(1-x^{2}\right)^{\frac{1}{2}} w(x) d x(1−x2)12w(x)dx that matter; w ( x ) w ( x ) w(x)w(x)w(x) the derivative of the absolutely continuous part of d α ( x ) d α ( x ) d alpha(x)d \alpha(x)dα(x).

R. Devore

  1. Let C ∗ [ − π , π ] C ∗ [ − π , π ] C^(**)[-pi,pi]C^{*}[-\pi, \pi]C∗[−π,π] denote the space of 2 π 2 π 2pi2 \pi2π-periodic continuous functions and ‖ ‖ ||\|‖. ‖ t h e s u p r e m u m n o r m o n [ − π , π ] ‖ t h e s u p r e m u m n o r m o n [ − π , π ] ||thesupremumnormon[-pi,pi]\| the supremum norm on [-\pi, \pi]‖thesupremumnormon[−π,π]. If ( L n L n L_(n)L_{n}Ln ) is a sequence of positive operators such that L n ( f ) L n ( f ) L_(n)(f)L_{n}(f)Ln(f) is a trigonometric polynomial of degree ≤ n ≤ n <= n\leq n≤n for
    each f f fff and n n nnn and if ( L n L n L_(n)L_{n}Ln ) satisfies the following conditions
( n → ∞ ) ‖ 1 − L n ( 1 ) ‖ = o ( 1 n 2 ) ‖ sin ⁡ x − L n ( sin ⁡ t , x ) ‖ = O ( 1 n 2 ) ‖ cos ⁡ x − L n ( cos ⁡ t , x ) ‖ = O ( 1 n 2 ) ( n → ∞ ) 1 − L n ( 1 ) = o 1 n 2 sin ⁡ x − L n ( sin ⁡ t , x ) = O 1 n 2 cos ⁡ x − L n ( cos ⁡ t , x ) = O 1 n 2 {:((n rarr oo)")"{:[||1-L_(n)(1)||=o((1)/(n^(2)))],[||sin x-L_(n)(sin t,x)||=O((1)/(n^(2)))],[||cos x-L_(n)(cos t,x)||=O((1)/(n^(2)))]:}:}\begin{array}{r} \left\|1-L_{n}(1)\right\|=o\left(\frac{1}{n^{2}}\right) \\ \left\|\sin x-L_{n}(\sin t, x)\right\|=O\left(\frac{1}{n^{2}}\right) \\ \left\|\cos x-L_{n}(\cos t, x)\right\|=O\left(\frac{1}{n^{2}}\right) \tag{$n\rightarrow\infty$} \end{array}(n→∞)‖1−Ln(1)‖=o(1n2)‖sin⁡x−Ln(sin⁡t,x)‖=O(1n2)‖cos⁡x−Ln(cos⁡t,x)‖=O(1n2)
then
( n → ∞ ) ‖ f − L n ( f ) ‖ = o ( 1 n 2 ) ( n → ∞ ) f − L n ( f ) = o 1 n 2 {:((n rarr oo)")"||f-L_(n)(f)||=o((1)/(n^(2))):}\begin{equation*} \left\|f-L_{n}(f)\right\|=o\left(\frac{\mathbf{1}}{n^{2}}\right) \tag{$n\rightarrow\infty$} \end{equation*}(n→∞)‖f−Ln(f)‖=o(1n2)
is equivalent to f f fff is constant on [ − π , π ] [ − π , π ] [-pi,pi][-\pi, \pi][−π,π].
2. Give an example of a sequence of positive operators ( L n L n L_(n)L_{n}Ln ) such that for each f ∈ C ( [ − 1 , 1 ] L n ( f ) ) f ∈ C [ − 1 , 1 ] L n ( f ) f in C([-1,1]L_(n)(f))f \in C\left([-1,1] L_{n}(f)\right)f∈C([−1,1]Ln(f)) is an algebraic polynomial of degree ≤ n ≤ n <= n\leq n≤n and
(i) ‖ 1 − L n ( 1 ) ‖ = o ( 1 n 2 ) (i) 1 − L n ( 1 ) = o 1 n 2 {:(i)||1-L_(n)(1)||=o((1)/(n^(2))):}\begin{equation*} \left\|1-L_{n}(1)\right\|=o\left(\frac{1}{n^{2}}\right) \tag{i} \end{equation*}(i)‖1−Ln(1)‖=o(1n2)
( n → ∞ ) ‖ x − L n ( t , x ) ‖ = O ( 1 n 2 ) ( n → ∞ ) x − L n ( t , x ) = O 1 n 2 {:((n rarr oo)")"||x-L_(n)(t,x)||=O((1)/(n^(2))):}\begin{equation*} \left\|x-L_{n}(t, x)\right\|=O\left(\frac{1}{n^{2}}\right) \tag{$n\rightarrow\infty$} \end{equation*}(n→∞)‖x−Ln(t,x)‖=O(1n2)
‖ x 2 − L n ( t 2 , x 2 ) ‖ = O ( 1 n 2 ) x 2 − L n t 2 , x 2 = O 1 n 2 ||x^(2)-L_(n)(t^(2),x^(2))||=O((1)/(n^(2)))\left\|x^{2}-L_{n}\left(t^{2}, x^{2}\right)\right\|=O\left(\frac{1}{n^{2}}\right)‖x2−Ln(t2,x2)‖=O(1n2)
where ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖ is the supremum norm on [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1][−1,1].

G. Freud

  1. We recently proved the following result: Let f ∈ Lip α , 0 < α < 1 f ∈ Lip α , 0 < α < 1 f in Lip alpha,0 < alpha < 1f \in \operatorname{Lip} \alpha, 0<\alpha<1f∈Lipα,0<α<1 a 2 π 2 π 2pi2 \pi2π-periodic and let
M 0 ( f ) = { x : f ( x + h ) − f ( x ) = o ( | h | α ) } M 1 ( f ) = { x : f ( x + h ) − f ( x − h ) = o ( | h | α ) } M 2 ( f ) = { x : f ( x + h ) + f ( x − h ) − 2 f ( x ) = o ( | h | α ) } M 0 ( f ) = x : f ( x + h ) − f ( x ) = o | h | α M 1 ( f ) = x : f ( x + h ) − f ( x − h ) = o | h | α M 2 ( f ) = x : f ( x + h ) + f ( x − h ) − 2 f ( x ) = o | h | α {:[M_(0)(f)={x:f(x+h)-f(x)=o(|h|^(alpha))}],[M_(1)(f)={x:f(x+h)-f(x-h)=o(|h|^(alpha))}],[M_(2)(f)={x:f(x+h)+f(x-h)-2f(x)=o(|h|^(alpha))}]:}\begin{aligned} & \mathfrak{M}_{0}(f)=\left\{x: f(x+h)-f(x)=o\left(|h|^{\alpha}\right)\right\} \\ & \mathfrak{M}_{1}(f)=\left\{x: f(x+h)-f(x-h)=o\left(|h|^{\alpha}\right)\right\} \\ & \mathfrak{M}_{2}(f)=\left\{x: f(x+h)+f(x-h)-2 f(x)=o\left(|h|^{\alpha}\right)\right\} \end{aligned}M0(f)={x:f(x+h)−f(x)=o(|h|α)}M1(f)={x:f(x+h)−f(x−h)=o(|h|α)}M2(f)={x:f(x+h)+f(x−h)−2f(x)=o(|h|α)}
further let f ~ f ~ tilde(f)\tilde{f}f~ be the harmonic conjugate of f f fff, so that by Privalov's theorem f ~ ∈ Lip α f ~ ∈ Lip α tilde(f)in Lip alpha\tilde{f} \in \operatorname{Lip} \alphaf~∈Lipα.
Theorem. Each two of the sets M 0 ( f ) ; M 1 ( f ) , M 2 ( f ) , M 0 ( f ~ ) , M 1 ( f ~ ) , M 2 ( f ~ ) M 0 ( f ) ; M 1 ( f ) , M 2 ( f ) , M 0 ( f ~ ) , M 1 ( f ~ ) , M 2 ( f ~ ) M_(0)(f);M_(1)(f),M_(2)(f),M_(0)( tilde(f)),M_(1)( tilde(f)),M_(2)( tilde(f))\mathfrak{M}_{0}(f) ; \mathfrak{M}_{1}(f), \mathfrak{M}_{2}(f), \mathfrak{M}_{0}(\tilde{f}), \mathfrak{M}_{1}(\tilde{f}), \mathfrak{M}_{2}(\tilde{f})M0(f);M1(f),M2(f),M0(f~),M1(f~),M2(f~) differ by a set of measure zero at most (G. Freud, Studia Mathematica, 1969). (Proved with the aid of trigonometric approximation.)
Problem 1 a a aaa : Extend the result to the sets
M k ( f ) = { x : Δ h k f ( x ) = o ( | h | α ) } M k ( f ) = x : Δ h k f ( x ) = o | h | α M_(k)(f)={x:Delta_(h)^(k)f(x)=o(|h|^(alpha))}\mathfrak{M}_{k}(f)=\left\{x: \Delta_{h}^{k} f(x)=o\left(|h|^{\alpha}\right)\right\}Mk(f)={x:Δhkf(x)=o(|h|α)}
Problem 1 b 1 b 1b1 b1b : The operator f → f ~ f → f ~ f rarr tilde(f)f \rightarrow \tilde{f}f→f~ is a special singular integral. Extend the result to singular integrals of more general type.
Problem 1c: Extend the result to functions of several variables.
2. Let e Q ( x ) e Q ( x ) e^(Q(x))\mathrm{e}^{Q(x)}eQ(x) be a weight function on the whole real axis, let us denote by Π n Π n Pi_(n)\Pi_{n}Πn the set of polynomials of degree at most n n nnn and for an arbitrary on ( − ∞ , + ∞ − ∞ , + ∞ -oo,+oo-\infty,+\infty−∞,+∞ ) continuous f f fff let
ε n ( Q ; f ) = inf p n ∈ Π n sup − ∞ < x < ∞ | f ( x ) − p n ( x ) | e − Q ( x ) . ε n ( Q ; f ) = inf p n ∈ Π n   sup − ∞ < x < ∞   f ( x ) − p n ( x ) e − Q ( x ) . epsi_(n)(Q;f)=i n f_(p_(n)inPi_(n))s u p_(-oo < x < oo)|f(x)-p_(n)(x)|e^(-Q(x)).\varepsilon_{n}(Q ; f)=\inf _{p_{n} \in \Pi_{n}} \sup _{-\infty<x<\infty}\left|f(x)-p_{n}(x)\right| \mathrm{e}^{-Q(x)} .εn(Q;f)=infpn∈Πnsup−∞<x<∞|f(x)−pn(x)|e−Q(x).
We recently proved that for a continuously differentiable f ( x ) f ( x ) f(x)f(x)f(x)
ε n ( x 2 ; f ) ≦ c n − 1 / 2 ε n ( x 2 ; f ′ ) ε n x 2 ; f ≦ c n − 1 / 2 ε n x 2 ; f ′ epsi_(n)(x^(2);f) <= cn^(-1//2)epsi_(n)(x^(2);f^('))\varepsilon_{n}\left(x^{2} ; f\right) \leqq c n^{-1 / 2} \varepsilon_{n}\left(x^{2} ; f^{\prime}\right)εn(x2;f)≦cn−1/2εn(x2;f′)
where n − 1 / 2 n − 1 / 2 n^(-1//2)n^{-1 / 2}n−1/2 is the best possible order of decrease. The emphasis of this result is on the fact, that no restriction concerning the rapidity of increase of f ( x ) f ( x ) f(x)f(x)f(x) for x → ± ∞ x → ± ∞ x rarr+-oox \rightarrow \pm \inftyx→±∞ is supposed.
Previous results on more general Q ( x ) Q ( x ) Q(x)Q(x)Q(x) from M. M. Dzrbasian (Dokl. A. N. USSR, 84 (1952), pp. 1123-1126) assume the uniform boundedness of f ( x ) f ( x ) f(x)f(x)f(x) and those of the speaker (G. Freud, Acta Math. Ac. Sci. Hung., 20 (1969), pp. 223-225) apply only for functions with polynomial growth.
Prove the more general inequality
ε n ( Q ; f ) ≦ α n ε n ( Q ; f ′ ) ε n ( Q ; f ) ≦ α n ε n Q ; f ′ epsi_(n)(Q;f) <= alpha_(n)epsi_(n)(Q;f^('))\varepsilon_{n}(Q ; f) \leqq \alpha_{n} \varepsilon_{n}\left(Q ; f^{\prime}\right)εn(Q;f)≦αnεn(Q;f′)
with correct order of α n α n alpha_(n)\alpha_{n}αn.

J. Musielak

  1. Let A C p , p ≥ 1 A C p , p ≥ 1 AC_(p),p >= 1A C_{p}, p \geq 1ACp,p≥1, be the Banach space of functions f f fff defined in the interval [ a , b ] , f ( a ) = 0 [ a , b ] , f ( a ) = 0 [a,b],f(a)=0[a, b], f(a)=0[a,b],f(a)=0, satisfying the following condition: for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 there exists a δ > 0 δ > 0 delta > 0\delta>0δ>0 such that for any finite system of non-overlapping subintervals ( a 1 , b 1 ) , … , ( a n , b n ) a 1 , b 1 , … , a n , b n (a_(1),b_(1)),dots,(a_(n),b_(n))\left(a_{1}, b_{1}\right), \ldots,\left(a_{n}, b_{n}\right)(a1,b1),…,(an,bn) of the interval [ a , b ] [ a , b ] [a,b][a, b][a,b] the inequality ∑ k = 1 n ( b k − a k ) p < δ ∑ k = 1 n   b k − a k p < δ sum_(k=1)^(n)(b_(k)-a_(k))^(p) < delta\sum_{k=1}^{n}\left(b_{k}-a_{k}\right)^{p}<\delta∑k=1n(bk−ak)p<δ implies ∑ k = 1 n | f ( b k ) − f ( a k ) | p < ε ∑ k = 1 n   f b k − f a k p < ε sum_(k=1)^(n)|f(b_(k))-f(a_(k))|^(p) < epsi\sum_{k=1}^{n}\left|f\left(b_{k}\right)-f\left(a_{k}\right)\right|^{p}<\varepsilon∑k=1n|f(bk)−f(ak)|p<ε, equipped with the norm ‖ f ‖ p = Var f ( x ) ‖ f ‖ p = Var f ( x ) ||f||_(p)=Var f(x)\|f\|_{p}=\operatorname{Var} f(x)‖f‖p=Varf(x). Let { B n ( f ) } B n ( f ) {B_(n)(f)}\left\{B_{n}(f)\right\}{Bn(f)} be the sequence of Bernstein polynomial of a function f ∈ A C p f ∈ A C p f in AC_(p)f \in A C_{p}f∈ACp. It is known that in case p = 1 , ‖ f − B n ( f ) ‖ 1 → 0 p = 1 , f − B n ( f ) 1 → 0 p=1,||f-B_(n)(f)||_(1)rarr0p=1,\left\|f-B_{n}(f)\right\|_{1} \rightarrow 0p=1,‖f−Bn(f)‖1→0 as n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞. Does the same hold for p > 1 p > 1 p > 1p>1p>1, i.e. does ‖ f − B n ( f ) ‖ p → 0 f − B n ( f ) p → 0 ||f-B_(n)(f)||_(p)rarr0\left\|f-B_{n}(f)\right\|_{p} \rightarrow 0‖f−Bn(f)‖p→0 as n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞ for any function f ∈ A C p f ∈ A C p f in AC_(p)f \in A C_{p}f∈ACp, where p > 1 p > 1 p > 1p>1p>1 ?
  2. Let C C CCC be the non-separable Banach space of uniformly almost periodic functions (in the sense of Bohr) on the real line, provided with the norm ‖ f ‖ C = sup − ∞ < x < ∞ | f ( x ) | ‖ f ‖ C = sup − ∞ < x < ∞   | f ( x ) | ||f||_(C)=s u p_(-oo < x < oo)|f(x)|\|f\|_{C}=\sup _{-\infty<x<\infty}|f(x)|‖f‖C=sup−∞<x<∞|f(x)|. Find an orthonormal Schauder basis in C C CCC.

J. Peetre

  1. Does the space C 1 C 1 C^(1)C^{1}C1 have the interpolation property with respect to the couple { C 0 , C 2 } C 0 , C 2 {C^(0),C^(2)}\left\{C^{0}, C^{2}\right\}{C0,C2}, i.e. is it true that
T : { C 0 , C 2 } → { C 0 , C 2 } ⇒ T : C 1 → C 1 ? T : C 0 , C 2 → C 0 , C 2 ⇒ T : C 1 → C 1 ? T:{C^(0),C^(2)}rarr{C^(0),C^(2)}=>T:C^(1)rarrC^(1)?T:\left\{C^{0}, C^{2}\right\} \rightarrow\left\{C^{0}, C^{2}\right\} \Rightarrow T: C^{1} \rightarrow C^{1} ?T:{C0,C2}→{C0,C2}⇒T:C1→C1?
It is known that this is true if we substitute Z Z ZZZ (Zygmund space) for C 1 C 1 C^(1)C^{1}C1 (cf. e.g. Lions-Peetre, Publ. Math. I.H.E.S. 19 (1964), 5-68). Also the corresponding result is known in the L p L p L_(p)L_{p}Lp-metric ( 1 < p < ∞ 1 < p < ∞ 1 < p < oo1<p<\infty1<p<∞ ), i.e. W p 1 W p 1 W_(p)^(1)W_{p}^{1}Wp1 has the interpolation property with respect to { L p , W p 2 } L p , W p 2 {L_(p),W_(p)^(2)}\left\{L_{p}, W_{p}^{2}\right\}{Lp,Wp2} (cf. Calderón, Studia Math. 24 (1964), 113-190). The proof depends however on the Marcinkiewicz multiplier theorem and does not generalize.
2. Let M p M p M_(p)M_{p}Mp be the space of Fourier-multipliers on L p L p L_(p)L_{p}Lp, i.e. a ∈ M p a ∈ M p a inM_(p)a \in M_{p}a∈Mp if and only if f ∈ L p ⇒ F − 1 f ∈ L p ⇒ F − 1 f inL_(p)=>F^(-1)f \in L_{p} \Rightarrow F^{-1}f∈Lp⇒F−1 a F f ∈ L p F f ∈ L p Ff inL_(p)F f \in L_{p}Ff∈Lp where F F FFF denotes the Fourier transform. Choose a "partition of unity" χ k ( k = 0 , ± 1 , ± 2 , … ) χ k ( k = 0 , ± 1 , ± 2 , … ) chi_(k)(k=0,+-1,+-2,dots)\chi_{k}(k=0, \pm 1, \pm 2, \ldots)χk(k=0,±1,±2,…) of the form χ k ( ξ ) == γ ( ξ / 2 k ) χ k ( ξ ) == γ ξ / 2 k chi_(k)(xi)==gamma(xi//2^(k))\chi_{k}(\xi)= =\gamma\left(\xi / 2^{k}\right)χk(ξ)==γ(ξ/2k) where γ γ gamma\gammaγ is a function whose support is contained in the interval ( 2 k − 1 , 2 k + 1 ) 2 k − 1 , 2 k + 1 (2^(k-1),2^(k+1))\left(2^{k-1}, 2^{k+1}\right)(2k−1,2k+1). It follows easily from a result of Hardy-LITTlewood (Quart (2 , 2 ). 10101 J. Math. 12 (1941), 221-256) that
( ∑ k = − ∞ ∞ ‖ χ k a ‖ M p q ) 1 / q < ∞ ⇒ a ∈ M P where 1 q = | 1 p − 1 2 | , 1 < p < ∞ . ∑ k = − ∞ ∞   χ k a M p q 1 / q < ∞ ⇒ a ∈ M P  where  1 q = 1 p − 1 2 , 1 < p < ∞ . (sum_(k=-oo)^(oo)||chi_(k)a||_(M_(p))^(q))^(1//q) < oo=>a inM_(P)quad" where "quad(1)/(q)=|(1)/(p)-(1)/(2)|,quad1 < p < oo.\left(\sum_{k=-\infty}^{\infty}\left\|\chi_{k} a\right\|_{M_{p}}^{q}\right)^{1 / q}<\infty \Rightarrow a \in M_{P} \quad \text { where } \quad \frac{1}{q}=\left|\frac{1}{p}-\frac{1}{2}\right|, \quad 1<p<\infty .(∑k=−∞∞‖χka‖Mpq)1/q<∞⇒a∈MP where 1q=|1p−12|,1<p<∞.
It is possible to replace q q qqq by a larger exponent? It follows from e.g. SteinZygmund, (Ann. Math. $ 5 $ 5 $5\$ 5$5 (1967), 337-349) that at least q = ∞ q = ∞ q=ooq=\inftyq=∞ is not enough.
3. Let E E EEE be a, say, finite dimensional vectorspace. Which set-functions f f fff satisfy
f ( M + N ) ≤ f ( M ) + f ( N ) f ( M + N ) ≤ f ( M ) + f ( N ) f(M+N) <= f(M)+f(N)f(M+N) \leq f(M)+f(N)f(M+N)≤f(M)+f(N)
where + denotes the Minkowsky sum (i.e. z ∈ M + N z ∈ M + N z in M+Nz \in M+Nz∈M+N if and only if z z zzz has a representation of the form z = x + y z = x + y z=x+yz=x+yz=x+y with x ∈ M x ∈ M x in Mx \in Mx∈M and y ∈ N ) y ∈ N ) y in N)y \in N)y∈N). I know two trivial solutions: 1 ∘ f ( M ) = log 2 1 ∘ f ( M ) = log 2 1^(@)f(M)=log_(2)1^{\circ} f(M)=\log _{2}1∘f(M)=log2 card M , 2 ∘ f ( M ) = dim M M , 2 ∘ f ( M ) = dim M M,2^(@)f(M)=dim MM, 2^{\circ} f(M)=\operatorname{dim} MM,2∘f(M)=dimM where - denotes the linear hull. They are closely connected with the notions of ε ε epsi\varepsilonε-entropy and n n nnn-dimensional width respectively, which explains my interest in the general case.
4. (cf. the second problem posed by Musielak). Does A C p A C p AC_(p)A C_{p}ACp have the interpolation property with respect to the couple { C , A C } { C , A C } {C,AC}\{C, A C\}{C,AC} ? A positive answer would in (cf. Besov, Trudy Mat. Inst. Steklov
60 (1961), 42-81)? It is known that (Peetre (unpublished); ct. also e.g. Krabbe, Math. Ann. 151 (1963), 219-238)
B p 1 / p , p ∩ C ⊂ A C p ⊂ B p 1 / p , ∞ B p 1 / p , p ∩ C ⊂ A C p ⊂ B p 1 / p , ∞ B_(p)^(1//p,p)nn C sub AC_(p)subB_(p)^(1//p,oo)B_{p}^{1 / p, p} \cap C \subset A C_{p} \subset B_{p}^{1 / p, \infty}Bp1/p,p∩C⊂ACp⊂Bp1/p,∞
but are the exponents q q qqq involved the best possible?

T. Popoviciu

Considérons un ensemble F F FFF du type I n I n I_(n)I_{n}In sur l'intervalle [ a , b ] [ a , b ] [a,b][a, b][a,b] et soit L ( x 1 , x 2 , … , x n ; f ∣ x ) L x 1 , x 2 , … , x n ; f ∣ x L(x_(1),x_(2),dots,x_(n);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)L(x1,x2,…,xn;f∣x) l'élément unique de F F FFF qui prend les mêmes valeurs que la fonction f ( x ) f ( x ) f(x)f(x)f(x) sur les noeuds x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}x1,x2,…,xn, supposés distincts.
Trouver tous les ensembles F F FFF tel que le quotient
f ( x n + 1 ) − L ( x 1 , x 2 , … , x n ; f ∣ x n + 1 ) g ( x n + 1 ) − L ( x 1 , x 2 , … , x n ; g ∣ x n + 1 ) f x n + 1 − L x 1 , x 2 , … , x n ; f ∣ x n + 1 g x n + 1 − L x 1 , x 2 , … , x n ; g ∣ x n + 1 (f(x_(n+1))-L(x_(1),x_(2),dots,x_(n);f∣x_(n+1)))/(g(x_(n+1))-L(x_(1),x_(2),dots,x_(n);g∣x_(n+1)))\frac{f\left(x_{n+1}\right)-L\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x_{n+1}\right)}{g\left(x_{n+1}\right)-L\left(x_{1}, x_{2}, \ldots, x_{n} ; g \mid x_{n+1}\right)}f(xn+1)−L(x1,x2,…,xn;f∣xn+1)g(xn+1)−L(x1,x2,…,xn;g∣xn+1)
soit une fonction symétrique des variables (distinctes) x 1 , x 2 , … , x n + 1 , g ( x ) x 1 , x 2 , … , x n + 1 , g ( x ) x_(1),x_(2),dots,x_(n+1),g(x)x_{1}, x_{2}, \ldots, x_{n+1}, g(x)x1,x2,…,xn+1,g(x) étant une fonction F F FFF-convexe ou F F FFF-concave donnée. La propriété doit avoir lieu pour toute fonction f ( x ) f ( x ) f(x)f(x)f(x) définie sur l'intervalle [ a , b ] [ a , b ] [a,b][a, b][a,b].
Pour la notion d'ensemble du type I n I n I_(n)I_{n}In (ensemble interpolatoire d'ordre n n nnn ) et pour les notions de fonction F F FFF-convexe et de fonction F F FFF-concave voir Elena Moldovan "Sur une généralisation des fonctions convexes", Mathematica, 1 (24), 1959, 49-80.
1972

Related Posts