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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.
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Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969)
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[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).
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PROPOSED PROBLEMS
R. Askey
- Let P_(n-1)(x)P_{n-1}(x) be a polynomial of degree n-1,x_(k,n)n-1, x_{k, n} the zeros of the polynomials orthogonal with respect to d alpha(x)d \alpha(x), a positive measure on [-1,1][-1,1], and lambda_(k)\lambda_{k} the corresponding Christoffel numbers. When is
[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}}
where AA is independent of nn and the polynomial P_(n-1)(x)P_{n-1}(x). For p=oop=\infty and p=2p=2 it holds for all positive measures with A=1A=1. It holds for 1 <= p <= oo1 \leqq p \leqq \infty if d alpha(x)=(1-x)^(alpha)(1+x)^(beta)dxd \alpha(x)=(1-x)^{\alpha}(1+x)^{\beta} d x for some values of (alpha,beta)(\alpha, \beta). For alpha=beta=-(1)/(2)\alpha=\beta=-\frac{1}{2} this was proven by Marcinkiewicz and I can prove it for alpha=beta > -(1)/(2)\alpha=\beta>-\frac{1}{2}; 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, k a positive integer.
2. Let AA and BB be two Banach spaces of functions on [-1,1][-1,1] with A sub BA \subset B. Let P_(n)(x)P_{n}(x) be a polynomial of degree nn. Assume that AA contains all polynomials of degree nn. Then
2. Let AA and BB be two Banach spaces of functions on [-1,1][-1,1] with A sub BA \subset B. Let P_(n)(x)P_{n}(x) be a polynomial of degree nn. Assume that AA contains all polynomials of degree nn. Then
||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}
Find the correct order of growth of a(n,A,B)a(n, A, B). My main interest is the case A=L^(q)(d alpha),B=L^(p)(d beta)A=L^{q}(d \alpha), B=L^{p}(d \beta), 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} 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 alpha(x)d \alpha(x) on [-1,1][-1,1] for which 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 fails for some continuous function for all p > 2p>2. (Askey) I conjecture that the measures associated with Pollaczek
polynomials have this property. These measures vanish so rapidly at x=+-1x= \pm 1 that 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}}} diverges, d alpha(x)=w(x)dxd \alpha(x)=w(x) d x.
4. Prove that 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 for all continuous functions if alpha >= beta > -1\alpha \geq \beta>-1 and
(i) if quad alpha > -(1)/(2)\quad \alpha>-\frac{1}{2} then 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)),
(ii) if -1 < alpha <= -(1)/(2)-1<\alpha \leqq-\frac{1}{2} then p < oo,gamma >= alpha,delta >= betap<\infty, \gamma \geqq \alpha, \delta \geqq \beta.
3. (Turán) Find a positive measure d alpha(x)d \alpha(x) on [-1,1][-1,1] for which 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 fails for some continuous function for all p > 2p>2. (Askey) I conjecture that the measures associated with Pollaczek
polynomials have this property. These measures vanish so rapidly at x=+-1x= \pm 1 that 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}}} diverges, d alpha(x)=w(x)dxd \alpha(x)=w(x) d x.
4. Prove that 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 for all continuous functions if alpha >= beta > -1\alpha \geq \beta>-1 and
(i) if quad alpha > -(1)/(2)\quad \alpha>-\frac{1}{2} then 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)),
(ii) if -1 < alpha <= -(1)/(2)-1<\alpha \leqq-\frac{1}{2} then p < oo,gamma >= alpha,delta >= betap<\infty, \gamma \geqq \alpha, \delta \geqq \beta.
This is known for 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=1. Condition (i) is best possible.
5. Let 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 0 and alpha+beta+1 >= 0\alpha+\beta+1 \geqq 0. Prove that the (C,alpha+beta+2)(C, \alpha+\beta+2) means are non-negative. For alpha=beta=-(1)/(2),alpha=beta=0\alpha=\beta=-\frac{1}{2}, \alpha=\beta=0, alpha=-beta=(1)/(2)\alpha=-\beta=\frac{1}{2} this was shown by Fejér, and for alpha=beta > -(1)/(2)\alpha=\beta>-\frac{1}{2} it was shown by Kogbetliantz.
6. Let d alpha(x)d \alpha(x) be a positive measure on [-1,1][-1,1] and p_(n)(x)p_{n}(x) the corresponding orthonormal polynomials. For which pp do we have 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, where S_(n)^(f)(x)S_{n}^{f}(x) is the nn-th partial sum of the orthogonal series in p_(n)(x)p_{n}(x). For
5. Let 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 0 and alpha+beta+1 >= 0\alpha+\beta+1 \geqq 0. Prove that the (C,alpha+beta+2)(C, \alpha+\beta+2) means are non-negative. For alpha=beta=-(1)/(2),alpha=beta=0\alpha=\beta=-\frac{1}{2}, \alpha=\beta=0, alpha=-beta=(1)/(2)\alpha=-\beta=\frac{1}{2} this was shown by Fejér, and for alpha=beta > -(1)/(2)\alpha=\beta>-\frac{1}{2} it was shown by Kogbetliantz.
6. Let d alpha(x)d \alpha(x) be a positive measure on [-1,1][-1,1] and p_(n)(x)p_{n}(x) the corresponding orthonormal polynomials. For which pp do we have 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, where S_(n)^(f)(x)S_{n}^{f}(x) is the nn-th partial sum of the orthogonal series in p_(n)(x)p_{n}(x). For
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 0
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}<1, I conjecture that the correct range is
{:[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}
Some case with d alpha(x)d \alpha(x) a set of point masses should be worked out to see if this influences the range of pp, or whether it is only the zeros of (1-x^(2))^((1)/(2))w(x)dx\left(1-x^{2}\right)^{\frac{1}{2}} w(x) d x that matter; w(x)w(x) the derivative of the absolutely continuous part of d alpha(x)d \alpha(x).
R. Devore
- Let C^(**)[-pi,pi]C^{*}[-\pi, \pi] denote the space of 2pi2 \pi-periodic continuous functions and ||\|.||thesupremumnormon[-pi,pi]\| the supremum norm on [-\pi, \pi]. If ( L_(n)L_{n} ) is a sequence of positive operators such that L_(n)(f)L_{n}(f) is a trigonometric polynomial of degree <= n\leq n for
each ff and nn and if ( L_(n)L_{n} ) satisfies the following conditions
{:((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}
then
{:((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*}
is equivalent to ff is constant on [-pi,pi][-\pi, \pi].
2. Give an example of a sequence of positive operators ( L_(n)L_{n} ) such that for each f in C([-1,1]L_(n)(f))f \in C\left([-1,1] L_{n}(f)\right) is an algebraic polynomial of degree <= n\leq n and
2. Give an example of a sequence of positive operators ( L_(n)L_{n} ) such that for each f in C([-1,1]L_(n)(f))f \in C\left([-1,1] L_{n}(f)\right) is an algebraic polynomial of degree <= n\leq n and
{:(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*}
{:((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*}
||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)
where ||*||\|\cdot\| is the supremum norm on [-1,1][-1,1].
G. Freud
- We recently proved the following result: Let f in Lip alpha,0 < alpha < 1f \in \operatorname{Lip} \alpha, 0<\alpha<1 a 2pi2 \pi-periodic and let
{:[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}
further let tilde(f)\tilde{f} be the harmonic conjugate of ff, so that by Privalov's theorem tilde(f)in Lip alpha\tilde{f} \in \operatorname{Lip} \alpha.
Theorem. Each two of the sets 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}) differ by a set of measure zero at most (G. Freud, Studia Mathematica, 1969). (Proved with the aid of trigonometric approximation.)
Problem 1 aa : Extend the result to the sets
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\}
Problem 1b1 b : The operator f rarr tilde(f)f \rightarrow \tilde{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))\mathrm{e}^{Q(x)} be a weight function on the whole real axis, let us denote by Pi_(n)\Pi_{n} the set of polynomials of degree at most nn and for an arbitrary on ( -oo,+oo-\infty,+\infty ) continuous ff let
2. Let e^(Q(x))\mathrm{e}^{Q(x)} be a weight function on the whole real axis, let us denote by Pi_(n)\Pi_{n} the set of polynomials of degree at most nn and for an arbitrary on ( -oo,+oo-\infty,+\infty ) continuous ff let
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)} .
We recently proved that for a continuously differentiable f(x)f(x)
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)
where 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) for x rarr+-oox \rightarrow \pm \infty is supposed.
Previous results on more general 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) 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
epsi_(n)(Q;f) <= alpha_(n)epsi_(n)(Q;f^('))\varepsilon_{n}(Q ; f) \leqq \alpha_{n} \varepsilon_{n}\left(Q ; f^{\prime}\right)
with correct order of alpha_(n)\alpha_{n}.
J. Musielak
- Let AC_(p),p >= 1A C_{p}, p \geq 1, be the Banach space of functions ff defined in the interval [a,b],f(a)=0[a, b], f(a)=0, satisfying the following condition: for every epsi > 0\varepsilon>0 there exists a delta > 0\delta>0 such that for any finite system of non-overlapping subintervals (a_(1),b_(1)),dots,(a_(n),b_(n))\left(a_{1}, b_{1}\right), \ldots,\left(a_{n}, b_{n}\right) of the interval [a,b][a, b] the inequality sum_(k=1)^(n)(b_(k)-a_(k))^(p) < delta\sum_{k=1}^{n}\left(b_{k}-a_{k}\right)^{p}<\delta implies 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, equipped with the norm ||f||_(p)=Var f(x)\|f\|_{p}=\operatorname{Var} f(x). Let {B_(n)(f)}\left\{B_{n}(f)\right\} be the sequence of Bernstein polynomial of a function f in AC_(p)f \in A C_{p}. It is known that in case p=1,||f-B_(n)(f)||_(1)rarr0p=1,\left\|f-B_{n}(f)\right\|_{1} \rightarrow 0 as n rarr oon \rightarrow \infty. Does the same hold for p > 1p>1, i.e. does ||f-B_(n)(f)||_(p)rarr0\left\|f-B_{n}(f)\right\|_{p} \rightarrow 0 as n rarr oon \rightarrow \infty for any function f in AC_(p)f \in A C_{p}, where p > 1p>1 ?
- Let CC 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)=s u p_(-oo < x < oo)|f(x)|\|f\|_{C}=\sup _{-\infty<x<\infty}|f(x)|. Find an orthonormal Schauder basis in CC.
J. Peetre
- Does the space C^(1)C^{1} have the interpolation property with respect to the couple {C^(0),C^(2)}\left\{C^{0}, C^{2}\right\}, i.e. is it true that
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} ?
It is known that this is true if we substitute ZZ (Zygmund space) for C^(1)C^{1} (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}-metric ( 1 < p < oo1<p<\infty ), i.e. W_(p)^(1)W_{p}^{1} has the interpolation property with respect to {L_(p),W_(p)^(2)}\left\{L_{p}, W_{p}^{2}\right\} (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} be the space of Fourier-multipliers on L_(p)L_{p}, i.e. a inM_(p)a \in M_{p} if and only if f inL_(p)=>F^(-1)f \in L_{p} \Rightarrow F^{-1} a Ff inL_(p)F f \in L_{p} where FF denotes the Fourier transform. Choose a "partition of unity" chi_(k)(k=0,+-1,+-2,dots)\chi_{k}(k=0, \pm 1, \pm 2, \ldots) of the form chi_(k)(xi)==gamma(xi//2^(k))\chi_{k}(\xi)= =\gamma\left(\xi / 2^{k}\right) where gamma\gamma is a function whose support is contained in the interval (2^(k-1),2^(k+1))\left(2^{k-1}, 2^{k+1}\right). It follows easily from a result of Hardy-LITTlewood (Quart (2 , 2 ). 10101 J. Math. 12 (1941), 221-256) that
2. Let M_(p)M_{p} be the space of Fourier-multipliers on L_(p)L_{p}, i.e. a inM_(p)a \in M_{p} if and only if f inL_(p)=>F^(-1)f \in L_{p} \Rightarrow F^{-1} a Ff inL_(p)F f \in L_{p} where FF denotes the Fourier transform. Choose a "partition of unity" chi_(k)(k=0,+-1,+-2,dots)\chi_{k}(k=0, \pm 1, \pm 2, \ldots) of the form chi_(k)(xi)==gamma(xi//2^(k))\chi_{k}(\xi)= =\gamma\left(\xi / 2^{k}\right) where gamma\gamma is a function whose support is contained in the interval (2^(k-1),2^(k+1))\left(2^{k-1}, 2^{k+1}\right). It follows easily from a result of Hardy-LITTlewood (Quart (2 , 2 ). 10101 J. Math. 12 (1941), 221-256) that
(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 .
It is possible to replace qq by a larger exponent? It follows from e.g. SteinZygmund, (Ann. Math. $5\$ 5 (1967), 337-349) that at least q=ooq=\infty is not enough.
3. Let EE be a, say, finite dimensional vectorspace. Which set-functions ff satisfy
3. Let EE be a, say, finite dimensional vectorspace. Which set-functions ff satisfy
f(M+N) <= f(M)+f(N)f(M+N) \leq f(M)+f(N)
where + denotes the Minkowsky sum (i.e. z in M+Nz \in M+N if and only if zz has a representation of the form z=x+yz=x+y with x in Mx \in M and y in N)y \in N). I know two trivial solutions: 1^(@)f(M)=log_(2)1^{\circ} f(M)=\log _{2} card M,2^(@)f(M)=dim MM, 2^{\circ} f(M)=\operatorname{dim} M where - denotes the linear hull. They are closely connected with the notions of epsi\varepsilon-entropy and nn-dimensional width respectively, which explains my interest in the general case.
4. (cf. the second problem posed by Musielak). Does AC_(p)A C_{p} have the interpolation property with respect to the couple {C,AC}\{C, A C\} ? 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)
4. (cf. the second problem posed by Musielak). Does AC_(p)A C_{p} have the interpolation property with respect to the couple {C,AC}\{C, A C\} ? 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)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}
but are the exponents qq involved the best possible?
T. Popoviciu
Considérons un ensemble FF du type I_(n)I_{n} sur l'intervalle [a,b][a, b] et soit L(x_(1),x_(2),dots,x_(n);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right) l'élément unique de FF qui prend les mêmes valeurs que la fonction f(x)f(x) sur les noeuds x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}, supposés distincts.
Trouver tous les ensembles FF tel que le quotient
(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)}
soit une fonction symétrique des variables (distinctes) x_(1),x_(2),dots,x_(n+1),g(x)x_{1}, x_{2}, \ldots, x_{n+1}, g(x) étant une fonction FF-convexe ou FF-concave donnée. La propriété doit avoir lieu pour toute fonction f(x)f(x) définie sur l'intervalle [a,b][a, b].
Pour la notion d'ensemble du type I_(n)I_{n} (ensemble interpolatoire d'ordre nn ) et pour les notions de fonction FF-convexe et de fonction FF-concave voir Elena Moldovan "Sur une généralisation des fonctions convexes", Mathematica, 1 (24), 1959, 49-80.