Introduction

We study the complexity of Banach space valued integration. The input data are assumed to be r-smooth. We consider both definite and indefinite integration and analyse the deterministic and the randomized setting. We develop algorithms, estimate their error, and prove lower bounds. In the randomized setting the optimal convergence rate turns out to be related to the geometry of the underlying Banach space.Then we study the corresponding problems for parameter dependent scalar integration. For this purpose we use the Banach space results and develop a multilevel scheme which connects Banach space and parametric case.

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“…For details and proofs we refer to [1] and [8]. Let X ⊗ Y be the algebraic tensor product of Banach spaces X and…”

Section: Preliminariesmentioning

confidence: 99%

Complexity of Banach Space Valued and Parametric Integration

Daun

Heinrich

2013

Springer Proceedings in Mathematics &Amp; Statistics

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“…At first we recall the basic notions of Grothendieck's metric theory of tensor products (cf., eg., [3], [4], [6], [9]), which will be used throughout this paper. A tensor norm α is a mapping which assigns to each pair (E, F ) of Banach spaces a norm α(·; E, F ) on the algebraic tensor product E ⊗ F (shorthand: E⊗ α F and E⊗ α F for the completion) such that (i) ε ≤ α ≤ π (ii) α satisfies the metric mapping property: If S ∈ L(E, G) and T ∈ L(F, H), then S ⊗ T : E ⊗ α F −→ G ⊗ α H ≤ S T Well-known examples are the injective tensor norm ε, which is the smallest one, and the projective tensor norm π, which is the largest one.…”

Section: On Tensor Norms and Associated Banach Idealsmentioning

confidence: 99%

“…In general it is a nontrivial subject to prove accessibility of maximal Banach ideals since non-accessibility can only appear on Banach spaces without the metric approximation property, and in 1992, Pisier made use of such a Banach space (the Pisier space P ) to construct a non-accessible maximal Banach ideal (cf. [3], 31.6). On the other hand, accessible Banach ideals allow a suggestive (algebraic) calculus which leads to further results concerning the local structure of operator ideals (e.g.…”

Section: Introductionmentioning

confidence: 98%

“…a transfer of the principle of local reflexivity from the operator norm to suitable ideal norms A (cf. [3], [11] and [12]). This paper is mainly devoted to the description of a large class of maximal injective Banach ideals which are totally accessible.…”

Section: Introductionmentioning

confidence: 99%

“…We will see a deep interplay between conjugates of Banach ideals, Hilbert space factorization, Grothendieck's inequality and tensor stable quasi-Banach ideals. We only deal with Banach spaces and most of our notations and definitions concerning Banach spaces and operator ideals are standard and can be found in the detailed monographs [3] and [13]. However, if (A, A) and (B, B) are given quasi-Banach ideals, we will use the shorter notation (A d , A d ) for the dual ideal (instead of (A dual , A dual )) and the abbreviation A 1 = B for the equality (A, A) = (B, B).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Local properties of accessible injective operator ideals

Oertel

1998

Czechoslovak Mathematical Journal

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In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a "good behaviour" of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of absolutely summing operators (prop. 4.1). By the famous Grothendieck inequality, every operator from L 1 to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization -via an operator version of Grothendieck's inequality (lemma 4.2). As a consequence, we obtain a class of injective ideals, which are "quasi-accessible", and with the help of tensor stability, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.

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Fourier type 2 operators with respect to locally compact abelian groups

Hinrichs¹,

Piotrowski²

2004

Mathematische Nachrichten

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A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that Tf 2 ≤ c f 2 holds for all X-valued functions f ∈ L X 2 (G) wheref is the Fourier transform of f . We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of R n .

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Introduction

Cited by 83 publications

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Complexity of Banach Space Valued and Parametric Integration

Complexity of Banach Space Valued and Parametric Integration

Local properties of accessible injective operator ideals

Fourier type 2 operators with respect to locally compact abelian groups

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