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This paper presents a method of reduction of measurement points in cylindrical near field using spline interpolation in complex domain to measure near field in high speed by using multi probe system.

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

Our understanding of the interplay between Poincare inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

The goal of this note is to consider the application of the complex interpolation space of Morrey spaces. Actually, the boundedness of Riesz potentials acting on Morrey spaces, which is obtained by Adams, is refined by means of the complex interpolation.

In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1\lt p\lt \infty$ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.

In Chapter 16 we give applications to various topics of Banach space theory, the most spectacular of which is a sharpened version of a celebrated result of Jean Bourgain on the Λ p problem.

The Butterfly lemma we present can be considered a reiteration theorem for differentials generated from a complex interpolation process for families of K\"othe spaces. The lemma will be used to clarify the effect of different configurations in the resulting differential (because although interpolation is an orientation-free process, the obtention of differentials is not) and to round off a few aspects of Kalton's interpolation theorem., Comment: This paper is to appear in Nonlinear Analisis - TMA

International audience; Multivariate Lagrange and Hermite interpolation are examples ofideal interpolation. More generally an ideal interpolation problemis defined by a set of linear forms, on the polynomial ring, whosekernels intersect into an ideal.For an ideal interpolation problem with symmetry, we addressthe simultaneous computation of a symmetry adapted basis of theleast interpolation space and the symmetry adapted H-basis ofthe ideal. Beside its manifest presence in the output, symmetry isexploited computationally at all stages of the algorithm.

We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.

A standard solution technique for linear operator equations of first kind is the Landweber scheme which is an iterative method that uses the negative gradient of the current residual as search direction, which is also called the Landweber direction. Though this method proves to be stable with respect to noisy data, it is known to be numerically slow for problems in Hilbert spaces and this behavior shows to be even worse in some Banach space settings. This is why the idea came up to use several search directions instead of the Landweber direction only which has led to the development of Sequential Subspace Optimization (SESOP) methods. This idea is related to the famous Conjugate Gradient (CG) techniques that are known to be amongst the most effective methods to solve linear equations in Hilbert spaces. Since CG methods decisively make use of the inner product structure, they have been inherently restricted to Hilbert spaces so far. SESOP methods in Banach spaces do not share the conjugacy property with CG methods. In this article we use the concept of generalized orthogonality in Banach spaces and apply metric projections to orthogonalize the current Landweber direction with respect to the search space of the last iteration. This leads to an accelerated SESOP method which is confirmed by various numerical experiments. Moreover, in Hilbert spaces our method coincides with the Conjugate Gradient Normal Error (CGNE) or Craig’s method applied to the normal equation. We prove weak convergence to the exact solution. Furthermore we perform a couple of numerical tests on a linear problem involving a random matrix and on the problem of 2D computerized tomography where we use different l p -spaces. In all experiments the orthogonalization of the search space shows superior convergence properties compared to standard SESOP. This especially holds for p close to 1. Letting p → 2 the more we recover the conjugacy property for the search directions and the more the convergence behaves independently of the size of the search space.

Often a given vector space has more than one natural norm. For example, the space of bounded continuous functions \(f:[0,1]\rightarrow \mathbb {C}\) can be equipped with the norms.

We study the interrelation between the limit $$L_p(\Omega )$$-Sobolev regularity $$\overline{s}_p$$ of (classes of) functions on bounded Lipschitz domains $$\Omega \subseteq \mathbb {R}^d$$, $$d\ge 2$$, and the limit regularity $$\overline{\alpha }_p$$ within the corresponding adaptivity scale of Besov spaces $$B^\alpha _{\tau ,\tau }(\Omega )$$, where $$1/\tau =\alpha /d+1/p$$ and $$\alpha >0$$ ($$p>1$$ fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for $$\overline{\alpha }_p$$ in terms of $$\overline{s}_p$$ simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of [11] . This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a GΓ-space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain Ω with data in the space of the integrable function with respect to the distance function to the boundary of Ω.

We show that the Kottman constant $K(\cdot)$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $X$. We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [CaGoPa17], namely that the isomorphic Kottman constant of a twisted-sum space is the maximum of the constants of the respective summands. Consequently, the Kalton--Peck space may be renormed to have Kottman's constant arbitrarily close to $\sqrt{2}$. For other classical parameters, such as the Whitley and the James constants, we prove the continuity with respect to the Kadets metric., 14 pp

The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions to the HJMM equation in a large class of function spaces, such as the weighted Lebesgue and Sobolev spaces. Thirdly, we study the ergodic properties of the solution to the HJMM equation. In particular, we analyse the Markov property of the solution and we find a sufficient condition for the existence and uniqueness of an invariant measure for the Markov semigroup associated to the HJMM equation (when the coefficients are time independent) in the weighted Lebesgue spaces.

The purpose of this paper is twofold. Firstly, the new matrix domains are constructed with the new infinite matrices and some properties are investigated. Furthermore, dual spaces of new matrix domains are computed and matrix transformations are characterized. Secondly, examples between new spaces with classical sequence spaces and sequence spaces which are derived by an infinite matrix are given in the table form., 15 pages, 6 tables. arXiv admin note: text overlap with arXiv:1611.06138

We prove various extensions of the Coifman–Rubio de Francia–Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓ r ( ℓ s ) {\ell ^{r}(\ell ^{s})} -boundedness, which implies R \mathcal {R} -boundedness in many cases. The proofs are based on new Littlewood–Paley–Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.

In this paper we obtain well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in $$L^p$$ -based Sobolev and Besov spaces in Lipschitz domains on a compact Riemannian manifold M of dimension $$m\ge 2$$ . For the mixed problem we refer to partially vanishing traces on Ahlfors regular sets. We exploit the continuity property of an operator related to the variational formulation of such a boundary value problem on complex interpolation scales of $$L^p$$ -based Sobolev spaces defined on M or on a Lipschitz domain of M, $$p\in (1,\infty )$$ , and the property that this operator is an isomorphism for $$p=2$$ . Then the stability of the quality of being isomorphism on complex interpolation scales leads to the extension of the well-posedness results of analyzed boundary value problems from $$p\!=\!2$$ to p in a neighborhood of 2. First, we focus on a variational approach that reduces boundary problems of transmission, Dirichlet and mixed type for the Brinkman system to equivalent mixed variational formulations with data in $$L^p$$ -based Sobolev and Besov spaces. For $$p=2$$ , such a mixed variational formulation is well-posed. The mixed variational formulation is further expressed in terms of a linear continuous operator on $$H^{1,q}\times L^q$$ -Sobolev spaces for any $$q\in (1,\infty )$$ , which is also invertible on the solution space corresponding to $$q=2$$ . Working on complex interpolation scales allows us to extend the invertibility of the operator for $$q=2$$ to a neighborhood of 2, and then to extend the well-posedness result to $$L^p$$ -based Sobolev spaces with p in a neighborhood of 2. Well-posedness results for the analyzed transmission problems allow us to define the layer potentials for the nonsmooth coefficient Brinkman system and to obtain their properties in $$L^p$$ -based Sobolev and Besov spaces. Then the solution of the Poisson problem of Dirichlet type is constructed explicitly in terms of such layer potentials. Finally, the Poisson problem of Neumann type is also analyzed and the corresponding well-posedness result in $$L^p$$ -based Sobolev and Besov spaces is also obtained. In addition, we determine the unique solution of the Neumann problem in the case $$p=2$$ , by using a layer potential approach. We extend the well-posedness results obtained in Kohr and Wendland (Boundary value problems for the Brinkman system with measurable coefficients in Lipschitz domains on compact Riemannian manifolds: a variational approach, 2018) for boundary problems for the nonsmooth coefficient Brinkman system in $$L^2$$ -based Sobolev spaces on Lipschitz domains in compact Riemannian manifolds to a more general setting of $$L^p$$ -based Sobolev and Besov spaces.

Generalized Stokes operators $$A_S$$ arise as linearizations of various models for non-Newtonian fluid flows. Here, it is proved that such operators in fairly general settings of domains and boundary conditions admit a bounded $${\mathcal H}^\infty $$ -calculus in the framework of solenoidal $$L_q$$ -spaces, $$1

In this work we study the following three space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation, and their splitting. We show that the answer to the three space problem is negative, giving two different solutions.

The Orlicz spaces $$X^{\varPhi }$$ associated to a quasi-Banach function space X are defined by replacing the role of the space $$L^1$$ by X in the classical construction of Orlicz spaces. Given a vector measure m, we can apply this construction to the spaces $$L^1_w(m),$$ $$L^1(m)$$ and $$L^1(\Vert m\Vert )$$ of integrable functions (in the weak, strong and Choquet sense, respectively) in order to obtain the known Orlicz spaces $$L^{\varPhi }_w(m)$$ and $$L^{\varPhi }(m)$$ and the new ones $$L^{\varPhi }(\Vert m\Vert ).$$ Therefore, we are providing a framework where dealing with different kind of Orlicz spaces in a unified way. Some applications to complex interpolation are also given.

We present a multiplier theorem on anisotropic Hardy spaces. When m satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator Tm: → , for the range of p that depends on the eccentricities of the dilation A and the level of regularity of a multiplier symbol m. This extends the classical multiplier theorem of Taibleson andWeiss.

We study complex interpolation of Herz‐type Triebel–Lizorkin spaces by using the Calderon product method. Additionally we present complex interpolation between Herz‐type Triebel–Lizorkin spaces and Triebel–Lizorkin spaces F∞,βs. Moreover, we apply these results to obtain the complex interpolation of Triebel–Lizorkin spaces equipped with power weights and between bmo (or hp) spaces and Herz spaces.

In this paper an anisotropic grand Sobolev–Morrey spaces are introduced. With the help of integral representation we study differential and differential-difference properties of functions from these spaces. Keywords: Anisotropic grand Sobolev–Morrey spaces, Integral representation, Embedding theorem, Hölder spaces

Let Ω be a bounded open subset of R n with a mild regularity property, let m ∈ N and p ∈ ( 1 , ∞ ) , and let W m , p ( Ω ) be the usual Sobolev space of order m based on L p ( Ω ) ; the closure in W m , p ( Ω ) of the smooth functions with compact support is denoted by W 0 m , p ( Ω ) . A special case of the results given below is that u ∈ W 0 m , p ( Ω ) if and only if all distributional derivatives of u of order m belong to L p ( Ω ) and u / d m ∈ L 1 ( Ω ) , where d ( x ) = dist ( x , ∂ Ω ) . In fact what is proved is the analogous result when the Sobolev space is based on a member of a class of Banach function spaces that includes both L p ( Ω ) and L p ( ⋅ ) ( Ω ) , the Lebesgue space with variable exponent p ( ⋅ ) satisfying natural conditions.

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolevtype spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a Poincare inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among other facts, the fact that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on quasi-Banach function lattices (and rearrangement-invariant spaces, in particular) are established and applied.

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on C 1 -domains. The coefficients are random functions depending on t , x and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain L p and Holder estimates of both the solution and its gradient.

This paper deals with extensions or twisted sums of Banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension. The results are presented in the context of interpolation of families of Banach spaces, and are applied to the study of submodules of Schatten classes. We also obtain nontrivial extensions of spaces without the CAP which also fail the CAP.

A mistake on a paper concerning near-vector spaces is fixed. A new characterization of near-vector spaces determined by finite fields is provided and the number (up to isomorphism) of these spaces is exhibited.

We give sufficient conditions for the best approximation of convex, bounded, closed and solid sets in Kothe–Bochner spaces and apply this result to sequence spaces.

We consider a Cauchy problem associated to a second-order evolution inclusion in non separable Banach spaces under Filippov type assumptions and we prove the existence of mild solutions.

We study conditions on Banach spaces close to \(L^1\) guaranteeing the existence of Random Unconditional Convergence and Divergence systems. Special attention is given to the Haar system and to Cesaro spaces.

This article presents necessary and sufficient optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem without constraints in terms of contingent derivatives in Banach spaces with stable functions. Using the steadiness and stability on a neighborhood of optimal point, necessary optimality conditions for efficient solutions are derived. Under suitable assumptions on generalized convexity, sufficient optimality conditions are established. Without assumptions on generalized convexity, a necessary and sufficient optimality condition for efficient solutions of unconstrained vector equilibrium problem is also given. Many examples to illustrate for the obtained results in the paper are derived as well.

We define a Hardy type space, by taking in the maximal characterization of Hardy spaces, the Wiener amalgam norms of the maximal functions, instead of the Lebesgue norms. The functions in this space can then behave differently locally and at infinity. We prove that this space contains the classical Hardy space and obtain an atomic decomposition.

We investigate order asymptotically isometric copies of c 0 , l ∞ and l 1 in Kothe spaces. Firstly we prove a number of general relations between them and we discuss what happens with such copies while passing to Kothe dual spaces, or vice versa – passing to preduals. Finally, we apply the results to characterize certain classes of Kothe spaces as Orlicz spaces, Calderon–Lozanovskiĭ spaces, Orlicz–Lorentz spaces and Musielak–Orlicz spaces with such copies.

We emphasize a bridge between two areas of function theory: hilbertian Muntz spaces and model spaces of the Hardy space of the right half plane. We give miscellaneous applications of this viewpoint to hilbertian Muntz spaces.

In this paper, the concept of a pairwise k -semi-stratifiable space is introduced and studied. Some characterizations of pairwise k -semi-stratifiable spaces by means of pairwise g -functions and semi-continuous functions are given. A new characterization of quasi-pseudo-metrizable spaces is obtained by using pairwise k -semi-stratifiable spaces, which improves a theorem of Marin in [17] .

A space X is a Collins–Roscoe space if it has a countable point network satisfying the Collins–Roscoe structuring mechanism. In this article, we introduce a new property on the subspaces of product spaces called the minimal continuous factorization (MCF) property. We prove that if a subspace Y of the topological product of a family of cosmic spaces has the countable MCF property, then C p ( Y ) has the Collins–Roscoe property, hence it is meta-Lindelof. We also prove that, if X is a σ-compact metric space, then C p ( X ) satisfies σ-finite (F).

We show that, under the continuum hypothesis, “to be universally separably injective” is not a 3-space property, as we wrongly claimed in the paper mentioned in the title.