Interpolating subspaces in approximation theory

Journal of Approximation Theory, 2001

We show among other things that if B is a linear space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, for which there is a continuous function h defined in a neighbourhood of 0 in the real line which is non-affine in every neighbourhood of 0 and satisfies |h(t)| [ k |t| for all t, such that h p b is in B whenever b is in B and the composite function is defined, then every function in C 0 (X) which can be approximated on every pair of points in X by functions in B can be approximated uniformly by functions in B.

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.

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.

1985

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.

Zeitschrift für Analysis und ihre Anwendungen, 2015

Let (Y 0 , Y 1) be a Banach couple and let X j be a closed complemented subspace of Y j , (j = 0, 1). We present several results for the general problem of finding necessary and sufficient conditions on the parameters (θ, q) such that the real interpolation space (X 0 , X 1) θ,q is a closed subspace of (Y 0 , Y 1) θ,q. In particular, we establish conditions which are necessary and sufficient for the equality (X 0 , X 1) θ,q = (Y 0 , Y 1) θ,q , with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that X 1 = Y 1 and X 0 is a subspace of codimension one in Y 0 .

Journal of Functional Analysis, 1986

We interpolate in the complex method some real-intermediate quasi-Banach spaces. This enables us for example to get in a unilied way complex interpolation of H,, spaces 0 <pO <pr 4 co, from the real interpolation results. The H, spaces could be the standard ones as well as weighted H, spaces, H,, spaces on product domains, etc. (' 1986 Academic Press. Inc An extension of the A. P. Calderon method of complex interpolation to quasi-Banach spaces was first considered by N. M. Riviere in [ 111. The main obstacle to a fully successful theory is the failure of the maximum principle for functions assuming values in a quasi-Banach space; see . We should point out that for the real interpolation method the situation is quite different; most central theorems from the Banach space setting hold in the quasi-Banach, and even in a more general situation.

Journal of Mathematical Analysis and Applications, 1991

Beauzamy in [l] proved that the Lions-Peetre spaces A,, (0 < 0 < 1, 1 <p < co) are reflexive if and only if the imbedding i: d(A) + E(A) is weakly compact. Later on, the Davis-Figiel-Johnson-Pelczynski factorization technique was again used in [lo] to prove that the operator T: A,, + Be,p is weakly compact if and only if T: d(A)-+ Z(g) is weakly compact, thereby generalizing the aforesaid result of Beauzamy. In this paper, a modification of this technique is used to give a characterization of the reflexivity for the more general spaces AH,t.

Proceedings of the American Mathematical Society, 2014

We show that multilinear interpolation can be lifted to multilinear operators from spaces generated by the minimal methods to spaces generated by the maximal methods of interpolation defined on a class of couples of compatible p-Banach spaces. We also prove mutlilinear interpolation theorem for operators on Calderón-Lozanovskii spaces between L p-spaces with 0 < p ≤ 1. As an application we obtain interpolation theorems for multilinear operators on quasi-Banach Orlicz spaces. We introduce relevant notation and we recall some definitions. Let (X, •) be a quasinormed space. A quasi-norm induces locally bounded topology. A complete quasi-normed space is called quasi-Banach space. If in addition we have for some 0 < p ≤ 1 x + y p ≤ x p + y p , x, y ∈ X,

Journal of Approximation Theory, 1973

Arkiv for Matematik, 1989

This originated the question of whether there are generalizations of Theorem L-P to the case Ao~A~ and BomB 1.

Arkiv för matematik, 1984

Given an interpolation couple (A0, A~), the approximation functional is dcfined

Acta Mathematica Scientia, 2021

Let M be a semifinite von Neumann algebra. We equip the associated noncommutative Lp-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for 1 < p < ∞ let be equipped with the operator space structure via real interpolation as defined by the second named author (J. Funct. Anal. 139 (1996), 500-539). We show that Lp,p(M) = Lp(M) completely isomorphically if and only if M is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for 1 < p < ∞ and 1 ≤ q ≤ ∞ with p = q L∞(M; ℓq), L 1 (M; ℓq) 1 p , p = Lp(M; ℓq) with equivalent norms, i.e., at the Banach space level if and only if M is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: for any finite sequence (x i ) ⊂ L + p (M), where 0 < r < q < ∞ and 0 < p ≤ ∞. If M is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if p ≥ r.

2015

Using the results in papers [2] and [3] in this paper we prove the existence of the interpolating spline-function by the null space dimension of operators A and T. 1. Introduction. Let X,Y,Z be Banach spaces. Suppose A is a bounded linear operator of X into Z and T is bounded linear operator of X into Y. The null space and the rang of operator A will be denoted by N (A) and R(A), respectively. Let R(A) = Z. For a fixed element z ∈ Z we write by

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.