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1997, Journal of Computational and Applied Mathematics
By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.
Journal of Approximation Theory, 1969
Journal of Approximation Theory, 1991
We study operators F from L, [ --K, x] into the space of trigonometric polynomials of degree m > n that satisfy II additional conditions OodFf) = g,(f), for all FE FH and all choices of points t,, . . . . t,,. 24
2007
The aim of this paper is to study many interpolation problems in the space of polynomials of w-degree n. In order to do this, some new results concerning the polynomial spaces of w-degree n are given. We consider only the case of functions in two variables, but all the results obtained can be easily extended to many variables. We found a set of conditions for which, Π n,w , the space of polynomials of w-degree is an interpolation space. More details are obtained for the weight w = (1, w 2 ).
Banach Journal of Mathematical Analysis, 2011
Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by Butzer and Scherer in 1968 and, independently, by Y. Brudnyi and N. Kruglyak in 1978, and popularized by Pietsch in his seminal paper of 1981. Pietsch was interested in the parallelism that exists between the theories of approximation spaces and interpolation spaces, so that he proved embedding, reiteration and representation results for approximation spaces. In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of approximation schemes (X, {A n }) and sequence spaces S, if S 1 ⊂ S 2 ⊂ c 0 (with strict inclusions) then the approximation space A(X, S 1 , {A n }) is properly contained into A(X, S 2 , {A n }). We also initiate a study of strict inclusions between interpolation spaces, for Petree's real interpolation method.
Journal of Mathematical Analysis and Applications, 2003
Using tensor products of Banach couples we study a class of interpolation functors with the property that to every Banach couple of Banach algebras they give an interpolation space which is a Banach algebra. For the real θ, 1-method we give a complete answer to the question of when the interpolation space is unital.
Journal of Approximation Theory, 2001
We study n-term wavelet-type approximations in Besov and Triebel-Lizorkin spaces. In particular, we characterize spaces of functions which have prescribed degree of n-term approximation in terms of interpolation spaces. These results are applied to identify interpolation spaces between Triebel-Lizorkin and Besov spaces.
arXiv (Cornell University), 2021
Our work is related to problems 73 and 74 of Mazur and Orlicz in "The Scottish Book" (ed. R. D. Mauldin). Let k 1 ,. .. , k n be nonnegative integers such that n i=1 k i = m, and let K(k 1 ,. .. , k n ; X), where K = R or C, be the smallest number satisfying the property: if L is any symmetric m-linear form on a Banach space X, then sup x i ≤1, i=1,2,...,n |L(x k1 1 ,. .. , x kn n)| ≤ K(k 1 ,. .. , k n ; X) sup x ≤1 |L(x,. .. , x)| , where the exponents k 1 ,. .. , k n , are as described above, and each k i denotes the number of coordinates in which the corresponding base variable appears. In the case of complex Banach spaces, the problem of optimising the constant C(k 1 ,. .. , k n ; X) is well-studied. In the more challenging case of real Banach spaces much less is known about the estimates for R(k 1 ,. .. , k n ; X). In this work, both real and complex settings are examined using results from the local theory of Banach spaces, as well as from interpolation theory of linear operators. In the particular case of complex L p (µ) spaces, and for certain values of p, our results are optimal. As an application, we prove Markov-type inequalities for homogeneous polynomials on Banach spaces.
2003
For the complex interpolation method, Kouba proved an important interpolation formula for tensor products of Banach spaces. We give a partial extension of this formula in the injective case for the Gustavsson-Peetre method of interpolation within the setting of Banach function spaces.
Mathematical theory and modeling, 2014
The aim of this article is to obtain the order of convergence of weighted space by interpolation polynomials on [-pi, pi]. More details can be found in the full paper.
Journal of Approximation Theory, 2008
We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2,. .. , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f)E n (f), n=1, 2,. .. , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f)= 0, where E n (f) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x,
Journal of Approximation Theory, 1999
2000
Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S <j>{r) dr,-" , VaC'Wo Y a (r) dr.
Journal of Approximation Theory, 1970
Advances in Mathematics, 1982
A detailed development is given of a theory of complex interpolation for families of Banach spaces which extends the well-known theory for pairs of spaces. 203 000 I-8708/82/030203-27$05.00/O Copyri%t 0 1982 by Academic Rew. Inc. All &hts of reproduction i n my fm resawd.
Publications of the Research Institute for Mathematical Sciences, 2001
Let E be a complex Banach space and F be a complex Banach algebra. We will be interested in the subspace IP f (n E, F) of P (n E, F) generated by the collection of functions ϕ n (n ∈ AE, ϕ ∈ L(E, F)) where ϕ n (x) = (ϕ(x)) n for each x ∈ E.
International Journal of Mathematics and Mathematical Sciences, 2011
A real Banach algebra of Newton interpolating series, used to approximate the solutions of multipoint boundary value problems for ODE's, is studied.
Arkiv för matematik, 1989
m-hikari.com
LetĀ = (A 1 , A 2 , • • • , A n) be a compatible n-tuple of Banach spaces. We may define the interpolation method in R n , and prove some related lemma and theorem.
Annali di Matematica Pura ed Applicata, 1982
In the present paper we study a general ]orm of Peetre's J-and K-methods o] interpolation. Special emphasis is given to the equivalence theorem for J-and K-spaces and to reiteration theorems.
Journal of Approximation Theory, 1996
A theory of best approximation with interpolatory contraints from a finitedimensional subspace M of a normed linear space X is developed. In particular, to each x # X, best approximations are sought from a subset M(x) of M which depends on the element x being approximated. It is shown that this``parametric approximation'' problem can be essentially reduced to the``usual'' one involving a certain fixed subspace M 0 of M. More detailed results can be obtained when (1) X is a Hilbert space, or (2) M is an``interpolating subspace'' of X (in the sense of [1]).
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