## What is/are Function Space?

Function Space - In this paper, we first construct the function space of ( L,M )-fuzzy Q-convergence spaces to show the Cartesian-closedness of the category ( L,M )- QC of ( L,M )-fuzzy Q-convergence spaces.^{[1]}Examples in function spaces are given.

^{[2]}In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in function spaces.

^{[3]}These properties provide us with necessary and sufficient conditions for an isomorphic classification of function spaces C p ⁎ ( X ) , where X is any countable metric space of scattered height less than or equal to ω.

^{[4]}The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space.

^{[5]}Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method.

^{[6]}Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry.

^{[7]}Also, some examples of function spaces satisfying the given conditions are considered.

^{[8]}In this paper we investigate weight cubature formula in function spaces of S.

^{[9]}We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.

^{[10]}In this work, the sequence-function space of glycoside hydrolase family 94 (GH94) was explored in detail, using a combined approach of phylogenetic analysis and sequence similarity networks.

^{[11]}) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty)$ such that $f^*\chi_{[0,1]}\in E$ and $f^*\chi_{[1,\infty)}\in L^2$.

^{[12]}The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.

^{[13]}We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain.

^{[14]}Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms.

^{[15]}It is an extensive study of metric spaces, including the core topics of completeness, compactness, and function spaces, with a good number of applications.

^{[16]}In this paper we incorporate derivatives of the Hamiltonian into the VQE and develop some hybrid quantum-classical algorithms, which explore both Hamiltonian and wave-function spaces for solving quantum chemistry problems more efficiently.

^{[17]}As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$.

^{[18]}A wavelet filter is used to acquire the informative matrix of each picture and decrease the dimensionality of the function space in the suggested method.

^{[19]}The first approach is universal, in the sense that it applies in general Banach spaces; the technical conditions are typically easy to verify in sequence spaces, but are more complicated in function spaces.

^{[20]}He was a mentor and supervisor to many other leading scientists who continued his quest to characterise structure and function space.

^{[21]}The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space.

^{[22]}Prospective statistical models developed on the datasets to efficiently explore the sequence-function space will guide towards the intelligent design of proteins and peptides through deep directed evolution.

^{[23]}The aim of this work is to show how the Splitting Lemma and the Representation Formula intrinsically determine a fiber bundle over the space of quaternionic slice regular functions and as a consequence, several properties of this function space are interpreted in terms of sections, pullbacks and isomorphism of fiber bundles.

^{[24]}In this paper, we investigate the strict inclusion relation associated with intersections and unions of a general family of function spaces.

^{[25]}By studying and solving the direct problem for the heat equation in composite materials, we can determine the function spaces and solve the inverse problem.

^{[26]}Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability.

^{[27]}The function space, initial and boundary conditions are carefully chosen so that it fixes the relative orientation and displacement, and we follow a gradient flow to let the interface find its optimal structure.

^{[28]}The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently.

^{[29]}We show that the positioning of the hydrogen-bonding sites, as well as their number, has a profound influence on the shape of the resulting ESF maps, revealing promising structure–function spaces for future experiments.

^{[30]}We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions.

^{[31]}Here, we establish all the function spaces, formalisms, and identities required to build a version of Mikusinski’s operational calculus which covers Caputo derivatives with respect to functions.

^{[32]}In this paper we survey the estimation of nonparametric regression models using wavelets, under different conditions on the innovations, on the predictor variables, on the function spaces involved and on the regularity conditions imposed.

^{[33]}The Function Space Optimization (FSO) method, recently developed by Feigl et al.

^{[34]}In the light of different time constants of the electrical power and natural gas systems, the continuous spatial-temporal optimal operation schedule model of IGPS is formulated in function space.

^{[35]}The low and intermediate modes are controlled by means of a Lyapunov function, while the high modes are controlled with operator estimates in function spaces based on the Wiener algebra.

^{[36]}We consider a semigroup acting on the function space L based a measure space.

^{[37]}We further show that this R{e}nyi variational formula holds over a range of function spaces; this leads to a formula for the optimizer under very weak assumptions and is also key in our development of a consistency theory for R{e}nyi divergence estimators.

^{[38]}Existence and stability of the resulting Gibbs posteriors are shown on function space under weak assumptions on the prior and model.

^{[39]}In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory.

^{[40]}In addition, we introduce Hajlasz–Sobolev spaces on G and show that the above operators are bounded on the above function spaces.

^{[41]}Examples of such sets include pointwise intervals, simultaneous bands, or balls in a function space, and they may be frequentist or Bayesian in interpretation.

^{[42]}Upon discretization, the model inherits several desirable attributes from this infinite-dimensional, function space viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions.

^{[43]}An intriguing feature of positive $$C_0$$ C 0 -semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups.

^{[44]}Organization study about traditional Balinese buildings layout in terms of space organization resulted in a large scale, function space, and circulation space.

^{[45]}Function Space Optimization (FSO) ist eine symbolische Regressionsmethode, die automatisiert Transferfunktionen aus Daten schätzen kann.

^{[46]}This new technique of using the three-line theorem enables us to extend the function spaces with ease.

^{[47]}The convergence of the wave takes place in the function spaces naturally related to the energy of the wave.

^{[48]}The introduced analytic - Bloch and - Besov type of functions with some interesting properties for these classes of function spaces are established within the constructions of their norms.

^{[49]}In order to solve and formulate this inverse problem, the function spaces must be defined and represented.

^{[50]}

## continuous time problem

The proposed continuous-time problem is solved in a finite-dimensional function space spanned by Bernstein polynomials, which converts the problem into a solvable mixed-integer linear programming problem.^{[1]}A function space-based method is developed to solve the proposed model, which converts the continuous-time problem into a mixed-integer linear programming problem with finite dimensional decision space.

^{[2]}A function space solution method is proposed to reduce the dimensionality of the continuous-time problem by modeling the parameter and decision trajectories in a function space formed by Bernstein polynomials, which converts the continuous-time problem into a linear programming problem.

^{[3]}A scalable and computationally efficient function space solution method is proposed that converts the continuous-time problems into mixed integer linear programming problems with finite-dimensional decision space.

^{[4]}

## measurable set valued

The first aim of this work is to introduce a new function space called the measurable set-valued Borel functions space and new topological space called the confine measurable set-valued Borel function topology on the set of the measurable set-valued Borel functions.^{[1]}

## Banach Function Space

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$.^{[1]}Let X be a ball Banach function space on R.

^{[2]}In this paper we extend the construction of Grand and Small Lebesgue spaces for the case of general Banach function spaces on finite measure space.

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^{[5]}We establish the boundedness of the Erdélyi-Kober fractional integral operators on ball Banach function spaces.

^{[6]}In the special case of Kothe-Bochner spaces of measurable vector-valued functions our main result asserts that every dominated operator T : E ( X ) → F from a Kothe-Bochner space E ( X ) to an order continuous Banach function space F has a unique representation T = T N + T H where T N is the narrow part and T H is the atomic part of T.

^{[7]}Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure.

^{[8]}Let Φ be a C∗-subalgebra of \(L^\infty (\mathbb {R})\) and \(SO_{X(\mathbb {R})}^\diamond \) be the Banach algebra of slowly oscillating Fourier multipliers on a Banach function space \(X(\mathbb {R})\).

^{[9]}Let $$\mathcal {M}_{X(\mathbb {R})}$$ be the Banach algebra of all Fourier multipliers on a Banach function space $$X(\mathbb {R})$$ such that the Hardy–Littlewood maximal operator is bounded on $$X(\mathbb {R})$$ and on its associate space $$X'(\mathbb {R})$$.

^{[10]}Let X be a ball quasi-Banach function space on Rn and HX(Rn) the associated Hardy space.

^{[11]}Recently, Leśnik [13] generalized this result to the setting of Toeplitz operators acting between abstract Hardy spaces H[X] and H[Y] built upon possibly different rearrangement-invariant Banach function spaces X and Y over $$\mathbb {T}$$ such that Y has nontrivial Boyd indices.

^{[12]}We then introduce the notion of convolution left-module action of $$L^1(G/H,\mu )$$ on the Banach function spaces $$L^p(G/H,\mu )$$.

^{[13]}

## Appropriate Function Space

These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power H α in the sense that similar estimates might fail with the classical Besov spaces.^{[1]}Appropriate function spaces and derivation of adjoint operators are investigated.

^{[2]}By using the semigroup theory, we first establish the existence of global weak and strong solutions as well as their continuous dependence on the initial data in appropriate function spaces, under suitable assumptions on the weight of time delay term, the external force term and the nonlinear term.

^{[3]}In this paper, the reconstruction problem of Acousto-Electric tomography is posed using the (smooth) complete electrode model, and a Levenberg-Marquardt iteration is formulated in appropriate function spaces.

^{[4]}Using the boundary integral equation method, we subsequently establish results for the existence of a solution in the appropriate function spaces.

^{[5]}To prove the main results of this paper, we establish sharp bounds for norms of appropriate function spaces.

^{[6]}One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model.

^{[7]}

## Modular Function Space

In this paper, we introduce the notion of ρ-attractive elements in modular function spaces.^{[1]}In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.

^{[2]}In this paper, we prove by means of a fixed-point theorem an existence result of the Cauchy problem associated to an ordinary differential equation in modular function spaces endowed with a reflexive convex digraph.

^{[3]}We use S -iteration to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the settings of modular function space.

^{[4]}The purpose of this paper is to introduce (λ, ρ)-quasi firmly nonexpansive mappings in context of modular function spaces and to prove some basic properties and approximation results for fixed point for these mappings.

^{[5]}In the present paper, we introduce the concept of relative weighted almost convergence and its weighted statistical extensions in multivariate modular function spaces based on a new type fractional-order double difference operator $$\nabla ^{a,b,c}_h$$∇ha,b,c.

^{[6]}

## Weighted Function Space

In these operators, σ not only features the sequences of operators but also features the Korovkin function set { 1 , σ , σ 2 } $\lbrace 1,\sigma ,\sigma ^{2} \rbrace $ in a weighted function space such that the operators fix exactly two functions from the set.^{[1]}Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate.

^{[2]}We construct the operators in a new q-Dunkl form and obtain the approximation properties in weighted function space.

^{[3]}In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type.

^{[4]}To find positive traveling solutions of the non-monotone system with a non-monotone incidence function, we will construct a suitable convex set in a weighted function space, and then apply Schauder fixed point theorem.

^{[5]}

## Valued Function Space

As applications, we obtain that the above operators are bounded on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces and on the vector-valued function spaces.^{[1]}The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting \`a la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Triebel-Lizorkin spaces.

^{[2]}We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra.

^{[3]}A factorization result is provided, and a specific representation of the biggest vector valued function space to which the operator can be extended is given.

^{[4]}

## Holomorphic Function Space

In this paper we study Toeplitz and Cesàro-type operators on holomorphic function spaces on a homogeneous Siegel domain of Type II.^{[1]}We study several connected problems of holomorphic function spaces on homogeneous Siegel domains.

^{[2]}Some weighted-type classes of holomorphic function spaces were introduced in the current study.

^{[3]}We construct a Hilbert holomorphic function space $H$ on the unit disk such that the polynomials are dense in $H$, but the odd polynomials are not dense in the odd functions in $H$.

^{[4]}

## Dimensional Function Space

The proposed continuous-time problem is solved in a finite-dimensional function space spanned by Bernstein polynomials, which converts the problem into a solvable mixed-integer linear programming problem.^{[1]}A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup.

^{[2]}For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space.

^{[3]}This is a linear operator acting in an infinite-dimensional function space which we can represent as a 2 × 2 matrix with operator entries.

^{[4]}

## Objective Function Space

One of the main advantages of our proposal are both its simplicity and its ability to scale up (in objective function space).^{[1]}In addition, this approach identifies a diverse set of solutions in the multi-objective function space which can be challenging to estimate with single-objective formulations.

^{[2]}Besides, the maximin strategy and crowding distance concept are employed for the selection of candidates to achieve the diversity and convergence of the solutions in the objective function space.

^{[3]}

## Variou Function Space

As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space $$A^{2}({\mathbb {D}})$$ , Fock space $$F^{2}({\mathbb {C}})$$ etc.^{[1]}The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces.

^{[2]}We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.

^{[3]}

## General Function Space

As an application, we give a characterization for the boundedness of the Volterra integral operator $$J_{g}$$ from $$\mathcal{L}_{p,\lambda}$$ to general function spaces $$F(p,p-1-\lambda,s)$$.^{[1]}Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces D p p−� to general function spaces are also investigated.

^{[2]}Meanwhile, the boundedness, compactness, and essential norm of the Volterra integral operator from Besov spaces to a class of general function spaces are also investigated.

^{[3]}

## Suitable Function Space

The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence.^{[1]}It is shown that the marginal function of the considered control system is lower semi-continuous and the optimal states operator generates a continuous branch in a suitable function space.

^{[2]}In this article, we shall follow the so-called Yau–Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space.

^{[3]}

## Different Function Space

Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces.^{[1]}In this paper, we obtain some inequalities about commutators of a rough p -adic fractional Hardy-type operator on Herz-type spaces when the symbol functions belong to two different function spaces.

^{[2]}Approximation properties in different function spaces are obtained, including quantitative Voronovskaya-type results.

^{[3]}

## Hilbert Function Space

Here, we present a bio-inspired global stochastic optimisation method applicable in Hilbert function spaces.^{[1]}The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

^{[2]}We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function $f(z)=1-\frac{1}{\sqrt{2}}(z_1+z_2)$ and a scale of Hilbert function spaces in the unit $2$-ball having reproducing kernel $(1-\langle z,w\rangle)^{-\gamma}$, $\gamma>0$.

^{[3]}

## Continuou Function Space

Using operator-valued Ċα -Fourier multiplier results on vector-valued Hölder continuous function spaces and the Carleman transform, we characterize the Cα -well-posedness of second order degenerate differential equations with infinite delay: (Mu)′′(t) = Au(t)+ ∫ t −∞ a(t− s)Au(s)ds+ f (t) and (Mu′)′(t) = Au(t)+ ∫ t −∞ a(t−s)Au(s)ds+ f (t) on R, where A : D(A)→ X and M : D(M)→ X are closed linear operators in a complex Banach space X , and a ∈ L(R+)∩L(R+; tα dt).^{[1]}In this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$ -Fourier multiplier results on vector-valued Holder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$ -well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$ , ( $t\in \mathbb{R}$ ), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$ , $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

^{[2]}

## Test Function Space

In the variational forms we introduce, the solution space is defined as a subspace V of the graph space associated with the differential operator in question, whereas the test function space L is a tuple of L 2 spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.^{[1]}Scalar and vector basis functions for the phase space \(\Omega \) (realizations of a turbulent flow) and the test function space \(\mathcal{N}_p\) (argument functions of the characteristic functional) plus analytic functions, for the purpose of testing numerically the convergence properties of the bases, are constructed using cylindrical coordinates suitable for the periodic flow through straight pipes with circular cross section.

^{[2]}

## Gevrey Function Space

We establish the well-posedness of the MHD boundary layer system in Gevrey function space without any structural assumption.^{[1]}We first establish the existence and uniqueness of solution in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ , then with the definition modulus of continuity, we show that the solution of Euler system is continuously dependent of the initial data $$v_{0}$$ in $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$.

^{[2]}

## Relevant Function Space

We construct the relevant function space, which is significantly constrained due to the extended Steinmann relations, up to weight 13 in coproduct form, and up to weight 12 as an explicit polylogarithmic representation.^{[1]}Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kreĭn-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

^{[2]}

## Proper Function Space

The proper function spaces and assumptions are proposed to discuss the existence of mild solutions.^{[1]}A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (

^{[2]}

## Certain Function Space

Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parametrization of a low-dimensional transition manifold in a certain function space.^{[1]}To unveil the inverse and composition properties of the IF operators, certain function spaces with their characterizations are presented and analyzed.

^{[2]}

## Analytic Function Space

In this current manuscript, some general classes of weighted analytic function spaces in a unit disc are defined and studied.^{[1]}In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces.

^{[2]}

## Normed Function Space

The Volterra operator is defined on an arbitrary normed function space F d that is continuously embedded in the space of square integrable functions defined on the unit d -cube.^{[1]}This paper will introduce you to some properties of normed function spaces with many groups variables field of Analysis and it helps me appreciate how normed Lebesgue–Morrey space with many groups of variables that build and studied new normed spaces nowadays.

^{[2]}

## Classical Function Space

This includes the basic definitions and properties of classical function spaces and distributions, the Fourier transform and the definition of Hadamard's finite part integrals which, in fact, represent the natural regularization of homogeneous distributions and of the hypersingular boundary integral operators.^{[1]}We characterize the reducing subspaces of S 2 ⁎ S 1 or S 1 S 2 ⁎ as wandering subspaces with additional structures, and give a unified way to describe the reducing subspaces of Toeplitz operators induced by non-analytic monomial on weighted Hardy spaces of several variables, which including many classical function spaces, such as weighted Bergman space and Dirichlet space over the polydisk.

^{[2]}