Search results for "Interpolation"

Some Classes of Operators on Partial Inner Product Spaces

2012

Many families of function spaces, such as $L^{p}$ spaces, Besov spaces, amalgam spaces or modulation spaces, exhibit the common feature of being indexed by one parameter (or more) which measures the behavior (regularity, decay properties) of particular functions. All these families of spaces are, or contain, scales or lattices of Banach spaces and constitute special cases of the so-called \emph{partial inner product spaces (\pip s)} that play a central role in analysis, in mathematical physics and in signal processing (e.g. wavelet or Gabor analysis). The basic idea for this structure is that such families should be taken as a whole and operators, bases, frames on them should be defined glo…

Error analysis for a special X-spline

1979

Clenshaw and Negus [1] defined the cubic X-spline, and they applied it to an interpolation problem. In the present paper, for the same interpolation problem, an interpolating splinew is considered by combining two specialX-splines. The construction ofw is such that the computational labour for its determination, in the case of piecewise equally spaced knots, is less than that of the conventional cubic splines c . A complete error analysis ofw is done. One of the main results is that, in the case of piecewise equally spaced knots,w ands c have essentially the same error estimates.

Preduals of spaces of homogeneous polynomials onLp-spaces

2012

Given a regular probability measure μ on a compact Hausdorff space, we explicitly describe the predual of the Banach space of continuous n-homogeneous polynomials on L p (μ) as the completion of a (explicit constructed) subspace of L p/n (μ) with respect to a (explicitly constructed) norm π p/n . An application to the factorization of dominated polynomials is provided.

Weighted Banach spaces of entire functions

1994

THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

2006

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l…

The Bishop–Phelps–Bollobás theorem for operators

2008

AbstractWe prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L1(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.

Existence of Unconditional Bases in Spaces of Polynomials and Holomorphic Functions

2002

Our main result shows that every Montel Kothe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space ((mE), τ0) of m-homegeneus polynomials on E endowed with the compact-open topology τ0 has an unconditional basis if and only if the space (ℋ(E), τδ) of holomorphic functions on E endowed with the bornological topology τδ associated to τ0 has an unconditional basis (for coechelon spaces τδ equals τ0). The main idea is to extend the concept of the Gordon-Lewis property from Banach to Frechet and (DF) spaces. In this way we obtain techniques which are used to characterize the existence of unconditional basis in spaces of m-th (symmetric) tenso…

(p,q)-summing sequences

2002

Abstract A sequence (x j ) in a Banach space X is (p,q) -summing if for any weakly q -summable sequence (x j ∗ ) in the dual space we get a p -summable sequence of scalars (x j ∗ (x j )) . We consider the spaces formed by these sequences, relating them to the theory of (p,q) -summing operators. We give a characterization of the case p=1 in terms of integral operators, and show how these spaces are relevant for a general question on Banach spaces and their duals, in connection with Grothendieck theorem.

Complemented Subspaces and Interpolation Properties in Spaces of Polynomials

1997

LetXbe a Banach space whose dualX* has typep ∈ (1, 2]. Ifmis an integer greater thanp/(p − 1) and (xn) is a seminormalized sequence weakly convergent to zero, there is a subsequence (yn) of (xn) such that, for each element (an) ofl∞, there is anm-homogeneous continuous polynomialPonXwithP(yn) = an,n = 1, 2,… . Some interpolation and complementation properties are also given in P(mlp), form < p, as well as in other spaces of polynomials and multilinear functionals.

On approximation of a class of stochastic integrals and interpolation

2004

Given a diffusion Y = (Y_{t})_{t \in [0,T]} we give different equivalent conditions so that a stochastic integral has an L 2-approximation rate of n −η, {\rm \eta \in (0,1/2],} if one approximates by integrals over piece-wise constant integrands where equidistant time nets of cardinality n + 1 are used. In particular, we obtain assertions in terms of smoothness properties of g(Y T ) in the sense of Malliavin calculus. After optimizing over non-equidistant time-nets of cardinality n + 1 in case {\rm \eta > 0} , it turns out that one always obtains a rate of n^{ - 1/2}, which is optimal. This applies to all functions g obtained in an appropriate way by the real interpolation method between th…