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This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

Suppose that X- = (X 0 , X 1 ) and Y-=(Y 0 ,Y 1 ) are Banach couples and suppose that T 0 : X 0 → Y 0 and T 1 : X 1 → Y 1 are bounded and linear. Also assume that Γ ∈ (Δ(X-))' and that T 0 and T 1 agree as maps from Δ(X-) ∩ ker Γ to Σ(Y-). If the maps do not agree as maps from all of Δ(X-) we cannot interpolate T 0 and T 1 to a map T : J 0,p (X- → J 0,p (Y-), where J 0,p denotes the classical J-method. This situation can for example be found in an article on interpolation of Hardy-type inequalities by Krugljak, Maligranda and Persson. We will in this paper define functors J 0,p;Γ such that T 0 and T 1 interpolate to a map T: J 0,p:Γ (X-) → J 0,p (Y-). The main purpose of this paper is to make the definition of the J 0,p;Γ (X-) spaces and build a theory for them. We will also do this for more general real parameters. If Γ is bounded on X 0 it holds that J 0,p:Γ (X-)=J 0,p (X 0 ∩ ker Γ, X 1 ). These spaces have been studied by Kalton, Ivanov and Lofstrom. Their results will follow as corollaries to the more general results of this article and our new theory can be thought of as a theory for generalized subcouples of codimension one. In the last section, we apply our theory to a situation considered by Krugljak, Maligranda and Persson in connection with Hardy-type inequalities. We prove new results and provide a new way of understanding that kind of problems.

It is known that if f ∈ W k p , then ω m ( f , t ) p ⩽ tω m −1 ( f ′, t ) p ⩽…. Its inverse with any constants independent of f is not true in general. Hu and Yu proved that the inverse holds true for splines S with equally spaced knots, thus ω m ( S , t ) p ∼ tω m −1 ( S ′, t ) p ∼ t 2 ω m −2 ( S ″, t ) p …. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.

Let B R ? ( h ? 2 ; f ) , ( R 0 ) denote the Bochner-Riesz means of order ? - 1 for the Dunkl transform of f ? L 1 ( R d ; h ? 2 d x ) associated with the weight function h ? 2 ( x ) : = ? j = 1 d | x j | 2 ? j on R d , where ? : = ( ? 1 , ? , ? d ) ? 0 , ∞ ) d . This paper shows that if ? ? 0 , then the Bochner-Riesz mean B R ? ( h ? 2 ; f ) ( x ) of each function f ? L 1 ( R d ; h ? 2 d x ) converges almost everywhere to f ( x ) on R d at the critical index ? = λ ? : = d - 1 2 + ? j = 1 d ? j as R ? ∞ . As is well-known in classical analysis, this result is no longer true in the unweighted case where ? = 0 , h ? ( x ) ? 1 , and B R ? ( h ? 2 ; f ) is the Bochner-Riesz mean of the Fourier transform.

In 1934 Kantorovitch modified the Bernstein polynomials Bn by means of metrical means to yield a nonlinear polynomial process B n ∗ which approximates measurable functions almost everywhere. The present paper is concerned with the pointwise comparison of B n and B n ∗ on C[0, 1] (the space of continuous functions on [0, 1]). We establish direct estimates of the form ¦(B n f − f)(x)¦ ⩽ ¦(B n ∗ f − f)(x)¦ + ω(3n − 1 3 , f) with the first modulus of continuity ω. On the other hand, it is the main purpose of this paper to show that this inequality can not be strengthened to ¦(B n f − f)(x)¦ ⩽ C x ¦(B n f − f)(x)¦ so that B n ∗ is not a pointwise extension of Bn on C[0, 1]. To this end, a previous condensation principle is applied concerning nonlinear functionals which assures the nonvalidity of this last inequality for x on a dense, in fact, residual, set of [0, 1].

In this case y,, is called a “best approximation” for x from A. If B is a sub- set of X, then 6(B, A)=sup{d(x, A);xeB}, is the deviation of B from A, and u!,(B, X)=inf{G(B, N); N’ IS an n-dimensional subspace of X} is the Kolmogrov n-width of B in X. If X and Y are normed linear spaces, then L(X, Y) denotes the set of all bounded linear operators from X to Y, K(X, Y) the set of all compact operators in L(X, Y) and K,(X, Y) the set of all operators in L(X, Y) of rank dn. The first serous study of the proximinality of K,(X, Y) in K(X, Y) and L(X, Y) appeared in the paper of Deutsch, Mach, and Saatkamp [2]. This paper was followed by two others, Kamal [4] and Kamal [S], in which several results concerning the proximinality of of K,(X, Y) in K(X, Y) and L(X, Y) were proved. In their paper [2], Deutsch et al. proved that for each integer IZ 2 0, the set K,(c,, c,) is proximinal in L(c,, c,), while in the present paper the following result is proved: Let Q and S be locally com

al. 121. iMore recently Tzimbalario [9] and Mohapatra and Sharma [4] have derived analogous results for certain classes of cardinal discrete splines. In this paper we derive results for a broad class of generalized cardinal splines which both unify and generalize the results of all the above papers. In particular we extend the results of [9] to cardinal Mermite interpolation. Let P, denote the space of all real-valued polynomials of degree not exceeding 12, and for 1 < s 9 n, let r = (yO , y1 ,.~., y8-J denote a set of linearly independent linear functionals on P, . We define Yn(r) to be the class of all functions S from [w to itself such that for v = 0, +l, &2,..., S 1 IV, u + 1) = S, E P, and y[S,-,(x + v)] = y[S,(x + Y)], Vy E F. If y&7) = p’i’(O), i = o,..., n - s, then Y,Z(r) is the class of cardinaji splines with integer nodes of multiplicity s studied ir, [2, 31. If Y&P) = @“j(O), i = o,..., r? - s - 1, and ~+~(p) = ~(‘L-s+l)(0), then Y.(S) consists of the cardinal g-splines studied by Lee and Sharma [I]. The cardinal discrete splines of [4, 9] are obtained by putting s = 1, y,(p) = p(iil2) and yi(p) = pci)(ih), i = 0, I,..., n - 1 (0 < h < 1 jiz). We note that our above definition of 5QF) is the analogy for cardinal po- lynomial splines of Schumakers’ classes of generalized splines as defined in [S]. Now if .cP = (pl , pz ,..., pJ is a basis for (p E P,: y(p) = 0, tl'y E f'), then it is easily seen that 5?Jr) comprises all functions S of the form S(x) = P(x) + i cyp

We consider the extremal problem of maximizing a point value � f ( z ) � at a given point z � G by some positive definite and continuous function f on a locally compact Abelian group (LCA group) G , where for a given symmetric open set � � z , f vanishes outside � and is normalized by f ( 0 ) = 1 .This extremal problem was investigated in R and R d and for � a 0-symmetric convex body in a paper of Boas and Kac in 1945. Arestov, Berdysheva and Berens extended the investigation to T d , where T : = R / Z . Kolountzakis and Revesz gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Caratheodory and Fejer.Moreover, following observations of Boas and Kac, Kolountzakis and Revesz showed how the general problem can be reduced to equivalent discrete problems of "Caratheodory-Fejer type" on Z or Z m : = Z / m Z . We extend their results to arbitrary LCA groups.

In this paper, the authors prove some Franke-Jawerth embedding for the Besov-type spaces B p , q s , ? ( R n ) and the Triebel-Lizorkin-type spaces F p , q s , ? ( R n ) . By using some limiting embedding properties of these spaces and the Besov-Morrey spaces N u , p , q s ( R n ) , the continuity envelopes in B p , q s , ? ( R n ) , F p , q s , ? ( R n ) and N u , p , q s ( R n ) are also worked out. As applications, the authors present some Hardy type inequalities in the scales of B p , q s , ? ( R n ) , F p , q s , ? ( R n ) and N u , p , q s ( R n ) , and also give the estimates for approximation numbers of the embeddings from B p , q s , ? ( ? ) , F p , q s , ? ( ? ) and N u , p , q s ( ? ) into C ( ? ) , where ? denotes the unit ball in R n .

The paper investigates the distribution of interpolation points of m 1 -maximally convergent multipoint Pade approximants with numerator degree ? n and denominator degree ? m n for meromorphic functions f on a compact set E ? C , where m n = o ( n / log n ) as n ? ∞ . It is shown that the normalized counting measures (resp. their associated balayage measures onto the boundary of E ) converge for a subsequence in the weak* sense to the equilibrium measure µ E of E if the multipoint Pade approximants for one single function f converge exactly in m 1 -measure on the maximal Green domain of meromorphy E ? ( f ) .

The aim of this paper is to prove a quantitative extension of Shapiro's result on the time-frequency concentration of orthonormal sequences in L α 2 ( R + ) . More precisely, we prove that, if { ? n } n = 0 + ∞ is an orthonormal sequence in L α 2 ( R + ) , then for all N ? 0 ? n = 0 N ( ? x ? n ? L α 2 2 + ? ? H α ( ? n ) ? L α 2 2 ) ? 2 ( N + 1 ) ( N + 1 + α ) , and the equality is attained for the sequence of Laguerre functions. Particularly if the elements of an orthonormal sequence and their Fourier-Bessel transforms (or Hankel transforms) have uniformly bounded dispersions then the sequence is finite.Moreover we prove the following strong uncertainty principle for bases for L α 2 ( R + ) , that is if { ? n } n = 0 + ∞ is an orthonormal basis for L α 2 ( R + ) and s 0 , then sup n ( ? x s ? n ? L α 2 2 ? ? s H α ( ? n ) ? L α 2 2 ) = + ∞ .

In this paper, we show that if E is an order continuous Kothe function space and Y is a separable subspace of X , then E ( Y ) is ball proximinal in E ( X ) if and only if Y is ball proximinal in X . As a consequence, E ( Y ) is proximinal in E ( X ) if and only if Y is proximinal in X . This solves an open problem of Bandyopadhyay, Lin and Rao. It is also shown that if E is a Banach lattice with a 1 -unconditional basis and for each n , Y n is a subspace of X n , then ( ⊕ Y n ) E is ball proximinal in ( ⊕ X n ) E if and only if each Y n is ball proximinal in X n .

A closed subspace Y of a normed linear space X is said to be ball proximinal if the closed unit ball of Y is proximinal in X. In this paper, we characterize ball proximinal and strongly ball proximinal hyperplanes.

This paper gives the estimates of the distance between two consecutive zeros of the nth m-orthogonal polynomial Pn for a Freud weight W=e−Q as follows. Let {xkn} be the zeros of Pn in decreasing order, an=an(Q) the nth Mhaskar–Rahmanov–Saff number, and ϕn(x)=max{n−2/3,1−|x|/an}. Assume that Q∈C(R) is even, Q(0)=0,Q′∈C[0,∞),Q′(x)>0,x∈(0,∞),Q″∈C(0,∞), and for some A,B>1, A≤(xQ′(x))′Q′(x)≤B,x∈(0,∞). Then, for 1≤k≤n−1, xkn−xk+1,n≤cannϕn(xkn)−1/2 and xkn−xk+1,n≥{cannϕn(xkn)−1/2,m=2,cannϕn(xkn)(m−2)/2,m≥3.Moreover, we have −an

We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is specialized to the energy, one obtains classical notions from Approximation Theory and Convex Analysis. The contribution of this paper is twofold. First, we provide an affirmative answer to a recently-posed question on whether or not every Klee set with respect to the right Bregman distance is a singleton. Second, we prove uniqueness of the Chebyshev center and we present a characterization that relates to previous works by Garkavi, by Klee, and by Nielsen and Nock., 23 pages, 2 figures, 14 images

In this paper, we investigate approximation of quasi-projection operators in Besov spaces B"p","q^@m, @m>0, [email protected]?p,[email protected]?~. Suppose I is a countable index set. Let (@f"i)"i"@?"I be a family of functions in L"p(R^s), and let (@[email protected]?"i)"i"@?"I be a family of functions in L"p"@?(R^s), where 1/p+1/[email protected]?=1. Let Q be the quasi-projection operator given by [email protected][email protected]?I@f"i,[email protected]?L"p(R^s). For h>0, by @s"h we denote the scaling operator given by @s"hf(x):=f(x/h), [email protected]?R^s. Let Q"h:[email protected]"[email protected]"1"/"h. Under some mild conditions on the functions @f"i and @[email protected]?"i ([email protected]?I), we establish the following result: If 0

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then–analogously to the Euclidean distance case–every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement the work by Hiriart-Urruty on the Euclidean case.

In this paper we show that a best monotone ϕ-approximant g on the cube (-1,1)n to a continuous and bounded function f is a best monotone ϕ-approximant on the level surfaces of g with respect to certain n-1-dimensional measure. This measure is mutually absolutely continuous with respect to the n-1-dimensional Haussdorff measure.

In this paper we give a lower bound for the strongly unique minimal projection (with norm one) constant (SUP-constant) onto some (n-k)-dimensional subspaces of l"~^n (n>=3, 1=

This paper studies numerical integration (or cubature) over the unit sphere S2 ⊂ R3 for functions in arbitrary Sobolev spaces Hs (S2), s > 1. We discuss sequences (Qm(n))n∈N of cubature rules, where (i) the rule Qm(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤ n and (ii) the sequence (Qm(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Qm(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space Hs (S2), s > 1, the worst-case cubature error has the order of convergence O(n-s), a result previously known only for the particular case s = 3/2. The crucial step in the extension to general s > 1 is a novel representation of ∑l=n+1∞(l + 1/2)-2s+1Pl(t), where Pl is the Legendre polynomial of degree l, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Qm(n). The order of convergence O(n-s) is optimal for sequences (Qm(n)) of cubature rules with properties (i) and (ii) if Qm(n) uses m(n) = O(n2) points.

In this paper, we show that the strong conical hull intersection property (CHIP) completely characterizes the best approximation to any x in a Hilbert space X from the set K : =C ∩ {x ∈ X :-g(x) ∈ S}, by a perturbation x - l of x from the set C for some l in a convex cone of X, where C is a closed convex subset of X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space and g : X → Y is a continuous S-convex function. The point l is chosen as the weak*-limit of a net of e-subgradients. We also establish limiting dual conditions characterizing the best approximation to any x in a Hilbert space X from the set K without the strong CHIP. The e-subdifferential calculus plays the key role in deriving the results.

Let Lq (1 ≤ q ≤ ∞) be the space of functions f measurable on I == [-1,1] and integrable to the power q, with norm ||f||q = { ∫-11 |f(x)|q dx}1/q. Li, is the space of functions measurable on I with norm ||f||r = esssup|x| ≤ 1 |f(x)| < ∞ We denote by AC the set of all functions absolutely continuous on I. For n ∈ N, q ∈ [1, ∞] we set Wn,q = {f : f(n-1) ∈ AC, f(n) ∈ Lq} In this paper, we consider the problem of accuracy of constants A, B in the inequalities ||f(m)||q ≤ A ||f||p + B||f(m-k+1)||r, m ∈ N, k ∈ W; p,q,r ∈ [1, ∞], f ∈ Wm-k-1.r.

In this paper we consider the problem of best approximation in lpn, 1 < p ≤ ∞. If hp, 1 < p < ∞, denotes the best lp-approximation of the element h ∈ Rn from a proper affine subspace K of Rn, h ∉ K, then limp→∞hp = h*∞ where h*∞ is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r ∈ N there are αj ∈ Rn, 1 ≤ j ≤ r, such that hp = h*∞ + α1/p-1 + α2/(p-1)2 + ... + αr/(p-1)r + γpr, with γp(r) ∈ Rn and ||γp(r)|| = O(p-r-1).

In this paper, we consider multivariate inhomogeneous refinement equations of the form @f(x)=@?"@a"@?"Z"^"sa(@a)@f(2x-@a)+g(x), x@?R^s, where @f=(@f"1, ..., @f"r)^T is the unknown, g=(g"1, ..., g"r)^T is a given vector of functions on R^s, and a is a finitely supported refinement mask such that each a(@a) is an rxr (complex) matrix. Let @f"0 be an initial vector of functions in the Sobolev space W^k"2(R^s). The corresponding cascade algorithm is given by @f"n(x)=@?"@a"@?"Z"^"sa(@a)@f"n"-"1(2x-@a)+g(x), x@?R^s, n=1, 2, .... A characterization is given for the strong convergence of the cascade algorithm in the Sobolev space W^k"2(R^s) (k@?N) in terms of the refinement mask a, the inhomogeneous term g, and the initial vector of functions @f"0.

In [J. Approx Theory71 (1992), 123-137], Al-Jarrah and Hasan considered the weight function W(x) = exp(−2 | x |α), α > 0, x ∈ R, and investigated the growth of an entire function ƒ that guarantees the geometric convergence of the Lagrange and Hermite interpolation processes and the Gauss-Jacobi quadrature formula for ƒ and its higher derivatives when the nodes of interpolation are chosen to be the zeros of the orthogonal polynomial associated with W. In this paper, we repeat investigations similar to those of Al-Jarrah and Hasan, but this time with a more general Freud-type weight function Ŵ2(x) = w2(x)exp(−2Q(x)), x ∈ R, where, for example, w(x) is a generalized Jacobi factor, and Q(x) is an even function that satisfies various restrictions.

In this paper, we introduce the concept of φ-average dimension far some subspaces of L[formula]( R n) and define the corresponding Kolmogorov φ-average ν-width of a set in L[formula]( R n). For the Sobolev class Wr[formula]( R n) in L[formula]( R n) we find necessary and sufficient conditions for this quantity to be finite and determine its asymptotic behaviour as ν → ∞. We also obtain the exact value of the average ν-widths of some classes of functions in L2( R n).

In this paper, we determine the exact value of average n − K width d n ( W r pq ( R ), L q ( R )) of Sobolev-Wiener class W r pq ( R ) in the metric L q ( R ) for 1 > q ≥ p > ∞ and get the value of d n ( W r p ( R ), L qp ( R )) for the dual case. We also solve the optimal interpolation problems of W r pq ( R ) in the metric L q ( R ) and W r p ( R ) in the metric L qp ( R ) for 1 q ≤ p

The subspaces Gα, Gβ, and Gβα (α, β ≥ 0)of Schwartz′ space S+ in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces Sα, Sβ, and Sβα are associated with the Fourier transform. Indeed, if we consider the Hankel transform Hγ (γ H γ(ƒ)(t) = 1 2 ∫∞0 (xt)−γ/2xγJγ([formula]) ƒ(x) dx then H γ is an isomorphism from Gα, Gβ, and Gβα onto Gα, Gβ, and Gαβ respectively. So. the spaces Gαα are invariant for H γ. In this paper, we characterize the spaces Gαα (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator F D defined by F D(ƒ)(w) = ∫∞0 ƒ(t) e−(1/2)((1 + w)/(1 − w))t for w ∈ D = {w ∈ C : |w| ≤ 1} when it acts on the space Gαα.

In this paper we show that the best approximation of a convex function by convex algebraic polynomials in L p , 1 ≤ p O (n −2/ p ).

Let X be a linear space, and H a Hilbert space. Let N denote a set of n distinct points in X designated by x1, ..., xn (these points are called nodes). It is desired to interpolate arbitrary data on N by a function in the linear span of the n functions, [formula] where yk are n distinct points in X (called knots), Tv are linear maps from X to H, and Fν are some suitable univariate functions. In this paper, we discuss the solvability of this interpolation scheme. For the case in which the nodes and knots coincide, we give a convenient condition which is equivalent to the nonsingularity of the interpolation matrices. We obtain some sufficient conditions for the case in which the nodes and knots do not necessarily coincide.

We investigate convergence in a weighted L ∞ -norm of Hermite-Fejer and Hermite interpolation and related approximation processes, when the interpolation points are zeros of orthogonal polynomials associated with weights W 2 = e −2 Q on the real line. For example, if H n (W 2 ,ƒ,x) denotes the n th Hermite-Fejer interpolation polynomial for W 2 = e −2 Q and the function ƒ, then we show that lim n→∞ xϵR { sup |H n (W 2 , f, x)−f(x)| W 2 (x)[1+|Q′(x)|] −k (1+|x|) −1 }=0 under suitable conditions on ƒ, W 2 , and κ. The weights to which the results are applicable include W 2 (x) = exp (−¦x¦ α ) , α > 1, or W 2 (x) = exp (− exp k (¦x¦ α )) , α > 1, k ⩾ 1, where exp k denotes the k th iterated exponential. Convergence of product integration rules induced by the various approximation processes is then deduced. Essentially the conclusion of the paper is that by damping the error in approximation of ƒ Hermite-Fejer or Hermite interpolation by a factor [1 + ¦Q′(x)¦] −κ (1 + ¦x¦) −1 , which decays much more slowly than the weight W 2 , we can ensure sup-norm convergence under quite general conditions.

Given a regular Norlund-method (N, p) one can prove that the sequence {σn(z)} of the Norlund-transforms of a power series ƒ(z) = ∑ v = 0 ∞ a v z v with radius of convergence r = 1 converges in at most countably many points outside the unit disc. In this paper we show that for a class of non-regular Norlund-methods the sequence {σn(z)} converges to an analytic function in a disc which strictly contains the unit disc, and the convergence is uniform on any compact subset of this disc.

LetΛ: 0 = λ0 < λ1λ < … be an infinite sequence of positive numbers, let n ϵ N and Bp(z): = Πk − 1n (z − λk − 1p)/(z + λk+1/p). Ganelius and Newman have shown that the expression εn(Λ)p = maxy ϵ R |Bp(1 + iy)| is the approximation index for the error μn(f, Λ)p: = infbk∥f(x) − ∑k = 0n bkxλk∥p of functions f ϵ Lp(0, 1) in the Lp-norm on [0, 1], 1 ⩽p⩽∞. That is, f is absolutely continuous on [0, 1], then μn(f, Λ)p⩽Apεn(Λ)p∥f ′∥p, where Ap < 229 is a numerical constant. It is the purpose of the present paper to apply another method of proof which produces small factors Ap < 42, 1 ⩽ p < 2. As is well-known, the factor Ap is small if 2 ⩽ p ⩽ ∞, for example, Ap < 14, which has been proved recently by the author.

If Φ(x) is defined on [−1, 1], let Ln(Φ, x) denote the Lagrange interpolation polynomial of degree n (or less) which agrees with Φ(x) at the equidistant nodes x k,n = −1 + (2k) n (k = 0, 1, …, n) . The classical Newton-Cotes integration formula approximates ∝−11 Φ(x) dx by ∝−11 Ln(Φ, x) dx. In this paper we present a very simple example of an analytic function Φ(x) for which limn → ∞ ∝−11 Ln(Φ, x) dx ≠ ∝−11 Φ(x) dx.

In a recent paper, L. Colzani studied the problem of monotone approximation of a function by its Fejer means. Of course, the approximation is monotone in L 2 (T), but he proved that this is no longer true in C(T). Here we show that it is hopeless to find a monotone approximation in C(T N ) whatever is the approximate unit we can use and we prove that results similar to that for the Fejer means hold for a large class of approximate units

Let B n (ƒ; P) be the n th Bernstein polynomial of a real function ƒ(P) whose domain is a triangle T . We show in this paper that if ƒ(P) is continuous on T and one of the inequalities B n (ƒ; P) ⩾ ƒ(P) or B n (ƒ; P) ⩾ B n + 1 (ƒ; P) holds for all positive integer n and all points of T , then ƒ cannot have a strict local maximum at an interior point of T .

The 1 1 2 -ball property and the strong 1 1 2 -ball property in a Banach space were studied by D. Yost [Bull. Austral. Math. Soc. 20 (1979) , 285–300; Math. Scand. 50 (1982) , 100–110]. G. Godini [“Banach Space Theory and Its Applications” (A. Pietsch, N. Popa and I. Singer, Eds.), Vol. 991, Springer-Verlag, New York/Berlin, 1983 ], gave geometrical characterizations of the subspaces with property (∗), as well as with the 1 1 2 -ball property. D. Yost [Math. Scand. 50 (1982) , 100–110] gave an example that has the 1 1 2 -ball property but not the strong 1 1 2 -ball property. In the present paper, property (S) is introduced and characterizations of the strong 1 1 2 -ball property are given. The subspaces of C(T) which have the 1 1 2 -ball property are characterized, where T is compact and connected.

Let V n = [ v 1 ,…, v n ] be the n -dimensional space of coordinate functions on a set of points ν⊂ R n , where v is the set of vertices of a regular convex polyhedron. In this paper it is demonstrated that the Fourier projection F is minimal from C( v ) onto V n . As a corollary, F is used to compute the absolute projection constant of any n -dimensional Banach space E n isometrically isomorphic to V n ⊂C( v ) , examples of which are the well-known cases E n = l n ∝ , l n 1 .

Let P(z) be a polynomial of degree n and P(z) its derivative. In this paper we shall obtain certain sharp generalization of some results of Govil, Malik and P. Tur¯ an concerning the maximum modulus of P(z) and P(z).

for all y in M. The set (perhaps empty) of all such elements m is denoted by 9,&x). Moreover, let a,,,, be the set of all x E X such that B,,,(x) # Qr. Clearly, we have a,,,, 1 M. We note that this kind of “approximation” has been introduced by Franchetti and Furi [S], and that several of its properties have been established in Refs. [8, 11, 123. Throughout this paper we shall assume that g is an increasing convex function defined on the interval [0, co) and such that g(0) = 0. An element m E M is said to be a strong coapproximation in M to an element XE X (with respect to g) if there exists a constant c = c(x) >O such that the inequality

The object of this paper is to study the rapidity of convergence of the Taylor mean of the Fourier series of ƒ(x) when ƒ(x) belongs to the class Lip(α, p). We show that it is of Jackson order provided that a suitable integrability condition is imposed upon the function ϑ x (t) = 1 2 {ƒ(x + t) − 2ƒ(x) + ƒ(x − t)} .

As is well known the Tchebycheff polynomial of degree n minimizes the sup norm over all monic polynomials with n simple zeros in [−1,+1). B. D. Bojanov [J. Approx. Theory, 26 (1979) , 293–300] recently investigated the situation for polynomials with a full set of zeros of higher multiplicities. In this paper we generalize these results to extended complete Tchebycheff systems.

In [9] a class of continued fractions called Laurent fractions were introduced, and it was shown that there is a one-to-one mapping between all Laurent fractions and all double sequences of real numbers {cn: n = 0, ±1, ±2, …} satisfying H2m(−2m) ≠ 0, H2m+1(−2m) ≠ 0, m = 0, 1, 2,…. (Hq(p) are the Hankel determinants associated with the double sequence.) The mapping is defined through the concept of correspondence between the continued fraction and the series ∑k=1∞ −c−kzk at z = 0 and the series ∑k=0∞ ckz−k at z = ∞. The subclass of contractive Laurent fractions is mapped onto those double sequences which satisfy H2m(−2m) > 0, H2m+1(−2m) > 0, m = 0, 1, 2,…. The main results of this paper are the following: (i) A sequence {Δk(z)} of discs connected with a contractive Laurent fraction is nested (for each z where Im z ≠ 0). (ii) The intersection ∩k = 1∞ Δk(z) is either a single point for every z or a closed disc for every z. (iii) General approximants Fk(z, τ) associated with a contractive Laurent fraction have partial fraction decompositions of the form F k (z, τ) = ∑ v = 1 n λ v (z + t v ) , where t v ϵ R , λ v > 0 .

In this paper, it is proven that the zeros of the Legendre polynomials Pn(x) satisfy the inequality (1 − xj − 1(n))(1 − xj + 1(n)) < (1 − xj(n))2, ∀jϵ {2, 3,…,n − 1}, ∀nϵ {3, 4,…}. This result is obtained by applying Sturm's comparison theorem to two homogeneous linear differential equations of second order, each of which has a particular solution deduced from the function [x(2 − x)]12Pn(1 − x), 0 ⩽ x ⩽ 2.

Let x be a real Banach space and (Ω, μ) a finite measure space. If φ is an increasing subadditive continuous function on [0, ∞) with φ(0) = 0, then we set Lφ(μ, X)= {f:Ω → X: ∥f ∥φ = ∝ φ(∥f (t)∥) dμ(t) < ∞}. One of the main results of this paper is: “For a closed subspace Y of X, Lφ(μ, Y) is proximinal in Lφ(μ, X) if and only if L1(μ, Y) is proximinal in L1(μ, X).” Hence if Y is a reflexive subspace of x, then Lp(μ, Y) is proximinal in Lp(μ, X) for all 0 < p < 1. Other results on proximinality of subsets of ιφ and Lφ(μ) are presented as well.

Let F j denote the linear functional that assigns to a real polynomial P n ( x ) = a 0 + a 1 x + a 2 x 2 + ··· + a n x n its j th partial coefficient sum F j ( P n ) = a 0 + a 1 + a 2 + ··· + a j (1 ⩽ j ⩽ n − 1; n ⩾ 5). It is demonstrated that a polynomial P n ∗ which is extremal for F j (i.e., ∥P n ∗ ∥ t8 = 1 (uniform norm on I = [ − 1, 1]) and F j (P n ∗ ) = ∥F j ∥) must have d alternation points on I , where n − 3 ⩽ d ⩽ n + 1. This result complements the author's previous one [ Math. Z. 182 (1983) , 549–552] stating that about; “half” the time, namely, if( j  n (mod 2), the n th Chebyshev polynomial of the first kind, ± T n , which posseses d = n + 1 alternation points on I , is extremal for F j . Known results on this subject are surveyed and additional topics such as uniqueness of polynomials with largest coefficient sums and practical estimation of ¦F j (P n )¦ are included, to make the paper self-contained.