Interpolation Spaces Research Articles

Reiteration theorem for [formula omitted] and [formula omitted]-spaces with the same parameter

Let E,F,E0,E1 be rearrangement invariant spaces; let a,b,b0,b1 be slowly varying functions and 0<θ0,θ1<1. We characterize the interpolation spaces(X‾θ0,b0,E0,a,FR,X‾θ1,b1,E1,a,FL)η,b,E,0≤η≤1, when the parameters θ0 and θ1 are equal (under appropriate conditions on bi(t), i=0,1). This completes the study started in [11,12,22], which only considered the case θ0<θ1. As an application we recover and generalize interpolation identities for grand and small Lebesgue spaces given by [26].

Complex interpolation of variable Morrey spaces

AbstractIn this paper, we investigate the description of complex interpolation between variable Morrey spaces. We show that the first complex interpolation between variable Morrey spaces equals to certain closed subspaces of variable Morrey spaces. In addition, we prove that the second complex interpolation of variable Morrey spaces coincides with their intermediate spaces. Our proof uses the characterization of the Calderón product between variable Morrey spaces. Our results cover complex interpolation of variable Lebesgue spaces and complex interpolation of (classical) Morrey spaces.

Notes on Real Interpolation of Operator Lp-Spaces

Let $${\cal M}$$ be a semifinite von Neumann algebra. We equip the associated non-commutative Lp-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for 1 < p < ∞ let $${L_{p,p}}({\cal M}) = {({L_\infty}({\cal M}),\,{L_1}({\cal M}))_{{1 \over p},p}}$$ 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 $${L_{p,p}}({\cal M}) = {L_p}({\cal M})$$ completely isomorphically if and only if $${\cal M}$$ 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 $${({L_\infty}({\cal M};{\ell _q}),\,\,{L_1}({\cal M};{\ell _q}))_{{1 \over p},p}} = {L_p}({\cal M};{\ell _q})$$ with equivalent norms, i.e., at the Banach space level if and only if $${\cal M}$$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $${\left\| {{{(\sum\limits_i {x_i^q})}^{{1 \over q}}}} \right\|_{{L_p}({\cal M})}} \le {\left\| {{{(\sum\limits_i {x_i^r})}^{{1 \over r}}}} \right\|_{{L_p}({\cal M})}}$$ for any finite sequence $$({x_i}) \subset L_p^ + ({\cal M})$$ , where 0 < r < q < ∞ and 0 < p ≤ ∞. If $${\cal M}$$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if p ≥ r.

Turbulence closure modeling with data-driven techniques: Investigation of generalizable deep neural networks

Generalizability of machine-learning (ML) based turbulence closures to accurately predict unseen practical flows remains an important challenge. It is well recognized that the neural network (NN) architecture and training protocol profoundly influence the generalizability characteristics. At the Reynolds-averaged Navier–Stokes level, NN–based turbulence closure modeling is rendered difficult due to two important reasons: inherent complexity of the constitutive relation arising from flow-dependent non-linearity and bifurcations; and, inordinate difficulty in obtaining high-fidelity data covering the entire parameter space of interest. Thus, a predictive turbulence model must be robust enough to perform reasonably outside the domain of training. In this context, the objective of the work is to investigate the approximation capabilities of standard moderate-sized fully connected NNs. We seek to systematically investigate the effects of (i) intrinsic complexity of the solution manifold; (ii) sampling procedure (interpolation vs extrapolation); and (iii) optimization procedure. To overcome the data acquisition challenges, three proxy-physics turbulence surrogates of different degrees of complexity (yet significantly simpler than turbulence physics) are employed to generate the parameter-to-solution maps. Lacking a strong theoretical basis for finding the globally optimal NN architecture and hyperparameters in the presence of non-linearity and bifurcations, a “brute-force” parameter-space sweep is performed to determine a locally optimal solution. Even for this simple proxy-physics system, it is demonstrated that feed-forward NNs require more degrees of freedom than the original proxy-physics model to accurately approximate the true model even when trained with data over the entire parameter space (interpolation). Additionally, if deep fully connected NNs are trained with data only from part of the parameter space (extrapolation), their approximation capability reduces considerably and it is not straightforward to find an optimal architecture. Overall, the findings provide a realistic perspective on the utility of ML turbulence closures for practical applications and identify areas for improvement.

Quartic Lp-convergence of cubic Riemannian splines

Abstract We prove quartic convergence of cubic spline interpolation for curves into Riemannian manifolds as the grid size of the interpolation grid tends to zero. In contrast to cubic spline interpolation in Euclidean space, where this result is classical, the interpolation operator is no longer linear. Still, concepts from the linear setting may be generalized to the Riemannian case, where we try to use intrinsic Riemannian formulations and avoid charts as much as possible.

The Butterfly lemma

The Butterfly lemma we present can be considered a reiteration theorem for differentials generated from a complex interpolation process for families of Köthe 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.

Interpolation and Extrapolation of Optimally Fitted Kinematic Error Model for Five-Axis Machine Tools

Abstract Machine tool geometric errors are frequently corrected by populating compensation tables that contain position-dependent offsets to each commanded axis position. While each offset can be determined by directly measuring the individual geometric error at that location, it is often more efficient to compute the compensation using a volumetric error model derived from measurements across the entire axis space. However, interpolation and extrapolation of measurements, once explicit in direct measurement methods, become implicit and obfuscated in the curve-fitting process of volumetric error methods. The drive to maximize model accuracy while minimizing measurement sets can lead to significant model errors in workspace regions at or beyond the range of the metrology equipment. In this paper, a method of constructing machine tool volumetric error models is presented in which the characteristics of the interpolation and extrapolation errors are constrained. Using a typical five-axis machine tool compensation methodology, a constraint bounding the tool tip modeled error slope is added to the error model identification process. By including this constraint over the entire space, the geometric errors over the interpolation space are still well identified. Also, the model performance over the extrapolation space is consistent with the behavior of the geometric error model over the interpolation space. The methodology is applied to an industrial five-axis machine tool. In the experimental implementation, for measurements outside of the measured region, an unconstrained model increases the mean residual by 40% while the constrained model reduces the mean residual by 40%.

Time periodic strong solutions to the Keller-Segel system coupled to Navier-Stokes equation

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N ( N ≥ 4 ) . Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C ( R , L s , ∞ ( R N ) ) when S ( n ) = n − 1 2 . Next for the case S ( n ) = 0 , we study the mild solution and point out it can become strong solution in B C ( R , L s , ∞ ( R N ) ) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations.

Opial properties in interpolation spaces

AbstractWe study interpolation spaces obtained via a general discrete interpolation method based on a Banach space with an unconditional basis. We find conditions which guarantee that such interpolation spaces have the Opial property and the uniform Opial property.

New interpolation spaces and strict H\xf6lder regularity for fractional abstract Cauchy problem

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

Navigating the amino acid sequence space between functional proteins using a deep learning framework.

MotivationShedding light on the relationships between protein sequences and functions is a challenging task with many implications in protein evolution, diseases understanding, and protein design. The protein sequence space mapping to specific functions is however hard to comprehend due to its complexity. Generative models help to decipher complex systems thanks to their abilities to learn and recreate data specificity. Applied to proteins, they can capture the sequence patterns associated with functions and point out important relationships between sequence positions. By learning these dependencies between sequences and functions, they can ultimately be used to generate new sequences and navigate through uncharted area of molecular evolution.ResultsThis study presents an Adversarial Auto-Encoder (AAE) approached, an unsupervised generative model, to generate new protein sequences. AAEs are tested on three protein families known for their multiple functions the sulfatase, the HUP and the TPP families. Clustering results on the encoded sequences from the latent space computed by AAEs display high level of homogeneity regarding the protein sequence functions. The study also reports and analyzes for the first time two sampling strategies based on latent space interpolation and latent space arithmetic to generate intermediate protein sequences sharing sequential properties of original sequences linked to known functional properties issued from different families and functions. Generated sequences by interpolation between latent space data points demonstrate the ability of the AAE to generalize and produce meaningful biological sequences from an evolutionary uncharted area of the biological sequence space. Finally, 3D structure models computed by comparative modelling using generated sequences and templates of different sub-families point out to the ability of the latent space arithmetic to successfully transfer protein sequence properties linked to function between different sub-families. All in all this study confirms the ability of deep learning frameworks to model biological complexity and bring new tools to explore amino acid sequence and functional spaces.

Wavelets and Real Interpolation of Besov Spaces

In view of the importance of Besov space in harmonic analysis, differential equations, and other fields, Jaak Peetre proposed to find a precise description of (Bp0s0,q0,Bp1s1,q1)θ,r. In this paper, we come to consider this problem by wavelets. We apply Meyer wavelets to characterize the real interpolation of homogeneous Besov spaces for the crucial index p and obtain a precise description of (B˙p0s,q,B˙p1s,q)θ,r.

Kinematic Tool-Path Smoothing for 6-Axis Industrial Machining Robots

The demands for machining by industrial robots have been increasing owing to their low installation cost and high flexibility. A novel trajectory generation algorithm for high-speed and high-accuracy machining by industrial robots is proposed in this paper. Linear interpolation in the workspace and smooth trajectory generation at the corners are important in industrial machining robots. Because industrial robots are composed of rotational joints, the joint space has a nonlinear relationship with the workspace. Therefore, linear interpolation in the joint space, which has been widely used in conventional machine tools, does not guarantee linear interpolation in the actual machining workspace. This results in the degradation of the machining surface. The proposed trajectory generation algorithm based on the decoupled approach can achieve linear interpolation in the workspace by separating the position commands into Cartesian coordinates and the orientation commands into spherical coordinates. In addition, a novel corner smoothing method that generates a smooth and continuous trajectory from discrete commands is proposed in this paper. The proposed kinematic local corner smoothing generates a smooth trajectory by using a 3-segmented constant jerk profile at the corners in the joint space. The sharp corners can thereby be replaced by smooth curves. The resulting cornering error is controlled by varying the cornering duration. The simulation results demonstrate the effectiveness of the proposed kinematic smoothing algorithm in achieving linear tool motion in straight sections and in generating smooth trajectories at corner sections within the user-defined tolerance.

Characterization of symmetrically $$\Delta $$-normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

Characterization of symmetrically $$\Delta $$-normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

Singular stochastic integral operators

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains.

Interpolation by Series of Exponential Functions Whose Exponents Are Condensed in a Certain Direction

For complex interpolation nodes, we study the problem of interpolation by series of exponential functions whose exponents form a set, which is condensed at infinity in a certain direction. We obtain a criterion for all sets of nodes from a special class. For arbitrary sets of nodes, we obtain a necessary condition for the solvability of a more general problem of interpolation by functions that can be represented as Radon integrals of an exponential function over a set of exponents. The paper also contains well-known results on interpolation, which, in particular, allow studying the multipoint holomorphic Vallée Poussin problem for convolution operators.

Parametric reduced order models based on a Riemannian barycentric interpolation

AbstractA new strategy for constructing parametric Galerkin reduced order models is presented in this article. This strategy is achieved thanks to the Riemannian manifold, quotient of the set of full‐rank matrices by the orthogonal group. Starting from a set of training parametrized subspaces of the same dimension, namely obtained by the proper orthogonal decomposition, the projection subspace for a new untrained parameter value is sought as the Karcher barycenter of the data points. The principal advantage of this strategy is that it leads to parametric reduced order models that are naturally flexible with respect to parameter variations. This property is a result of the simple expressions of the geodesic exponential and logarithmic mappings involved in the calculations of the approximated projection subspace. Confrontation with the usual interpolation in the tangent space to the Grassmann manifold is carried out for the flow problem past a circular cylinder and the flow in a lid‐driven cavity. Comparable results in terms of accuracy are obtained, while the proposed approach is shown to be computationally cheaper by allowing real‐time update of Galerkin projections.

Real interpolation of martingale Orlicz Hardy spaces and BMO spaces

In this article, the authors prove that the real interpolation spaces between martingale Orlicz Hardy spaces and martingale BMO spaces are martingale Orlicz–Lorentz Hardy spaces. Using sharp maximal functions, the authors also establish the characterizations of martingale Orlicz Hardy spaces.

Fractional Diffusion-Wave Equations: Hidden Regularity for Weak Solutions

Abstract We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

Some Reiteration Theorems for R , L , R R , R L , L R , and L L Limiting Interpolation Spaces

We consider the K -interpolation methods involving slowly varying functions. We establish some reiteration formulae including so-called L or R limiting interpolation spaces as well as the R R , R L , L R , and L L extremal interpolation spaces. These spaces arise in the limiting situations. The proofs of most reiteration formulae are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces in critical cases are given.