Approximation by elements of a finite-dimensional subspace of functions from various sobolev or nikol'skii spaces

1988 ◽  
Vol 43 (6) ◽  
pp. 444-454 ◽  
Author(s):  
V. N. Temlyakov
Keyword(s):  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Finite Dimensional

scholarly journals Functional ARCH and GARCH Models: A Yule-Walker Approach

2020 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Financial Time Series ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Stationary Solutions ◽  
Finite Dimensional Subspace ◽  
Estimation Errors ◽  
Linear Processes ◽  
Finite Dimensional ◽  
Garch Processes ◽  
Asymptotic Upper Bound

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

Implementing Galerkin Finite Element Methods for Semilinear Elliptic Differential Inclusions

2013 ◽  
Vol 13 (1) ◽  
pp. 95-118 ◽  
Author(s):  
Janosch Rieger
Keyword(s):  
Finite Element ◽  
Differential Inclusions ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Basic Algorithm ◽  
Galerkin Finite Element ◽  
Finite Dimensional ◽  
Partial Differential Inclusions ◽  
Galerkin Finite Element Methods ◽  
Element Approach

Abstract. This paper presents the first feasible method for the approximation of solution sets of semi-linear elliptic partial differential inclusions. It is based on a new Galerkin Finite Element approach that projects the original differential inclusion to a finite-dimensional subspace of . The problem that remains is to discretize the unknown solution set of the resulting finite-dimensional algebraic inclusion in such a way that efficient algorithms for its computation can be designed and error estimates can be proved. One such discretization and the corresponding basic algorithm are presented along with several enhancements, and the algorithm is applied to two model problems.

scholarly journals Quasi-reliable estimates of effective sample size

2020 ◽  
Author(s):  
Youhan Fang ◽  
Yudong Cao ◽  
Robert D Skeel
Keyword(s):  
Sample Size ◽  
Dimensional Subspace ◽  
Monte Carlo Algorithm ◽  
Finite Dimensional Subspace ◽  
Effective Sample Size ◽  
Finite Dimensional ◽  
Adequate Sampling ◽  
Autocorrelation Time ◽  
The Mean ◽  
The Cost

Abstract The efficiency of a Markov chain Monte Carlo algorithm for estimating the mean of a function of interest might be measured by the cost of generating one independent sample, or equivalently, the total cost divided by the effective sample size, defined in terms of the integrated autocorrelation time. To ensure the reliability of such an estimate, it is suggested that there be an adequate sampling of state space— to the extent that this can be determined from the available samples. A sufficient condition for adequate sampling is derived in terms of the supremum of all possible integrated autocorrelation times, which leads to a more stringent condition for adequate sampling than that simply obtained from integrated autocorrelation times for functions of interest. A method for estimating the supremum of all integrated autocorrelation times, based on approximation in a finite-dimensional subspace, is derived and evaluated empirically.

scholarly journals Characterizations of Operator Birkhoff–James Orthogonality

2017 ◽  
Vol 60 (4) ◽  
pp. 816-829 ◽  
Author(s):  
Mohammad Sal Moslehian ◽  
Ali Zamani
Keyword(s):  
Unit Vector ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Inner Product ◽  
Finite Dimensional Subspace ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
James Orthogonality ◽  
New Type ◽  
Norm Attaining

AbstractIn this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C*-modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , T is the strong Birkhoff–James orthogonal to S if and only if there exists a unit vector such that . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product C*-modules.

scholarly journals CALDERÓN’S INVERSE PROBLEM WITH A FINITE NUMBER OF MEASUREMENTS

2019 ◽  
Vol 7 ◽  
Author(s):  
GIOVANNI S. ALBERTI ◽  
MATTEO SANTACESARIA
Keyword(s):  
Finite Number ◽  
Reconstruction Algorithm ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Stability Estimates ◽  
Inverse Conductivity Problem ◽  
Finite Dimensional ◽  
First Results ◽  
Boundary Measurements ◽  
Inverse Boundary Value Problems

We prove that an $L^{\infty }$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace ${\mathcal{W}}$ . As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces ${\mathcal{W}}$ , including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim {\mathcal{W}}$ .

scholarly journals Galerkin approximation of linear problems in Banach and Hilbert spaces

2020 ◽  
Author(s):  
W Arendt ◽  
I Chalendar ◽  
R Eymard
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Galerkin Approximation ◽  
Dimensional Subspace ◽  
Adjoint Problem ◽  
Finite Dimensional Subspace ◽  
Finite Dimensional ◽  
Regularity Properties ◽  
Sesquilinear Form

Abstract In this paper we study the conforming Galerkin approximation of the problem: find $u\in{{\mathcal{U}}}$ such that $a(u,v) = \langle L, v \rangle $ for all $v\in{{\mathcal{V}}}$, where ${{\mathcal{U}}}$ and ${{\mathcal{V}}}$ are Hilbert or Banach spaces, $a$ is a continuous bilinear or sesquilinear form and $L\in{{\mathcal{V}}}^{\prime}$ a given data. The approximate solution is sought in a finite-dimensional subspace of ${{\mathcal{U}}}$, and test functions are taken in a finite-dimensional subspace of ${{\mathcal{V}}}$. We provide a necessary and sufficient condition on the form $a$ for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that ${{\mathcal{U}}}$ has a finite-dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case a generalization of the Aubin–Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side $L$, as shown by several applications. Finally, a section entitled ‘Supplement’ provides some consequences of our results for the approximation of saddle point problems.

scholarly journals Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 941 ◽  
Author(s):  
Milton Ferreira ◽  
Teerapong Suksumran
Keyword(s):  
General Theory ◽  
Decomposition Theorem ◽  
Dimensional Subspace ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Fiber Bundles ◽  
Finite Dimensional Subspace ◽  
Isomorphism Theorem ◽  
Quotient Spaces ◽  
Finite Dimensional

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.

scholarly journals Functional ARCH and GARCH Models: A Yule-Walker Approach

2019 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Financial Time Series ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Stationary Solutions ◽  
Finite Dimensional Subspace ◽  
Estimation Errors ◽  
Linear Processes ◽  
Finite Dimensional ◽  
Garch Processes ◽  
Asymptotic Upper Bound

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound for the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

scholarly journals Functional ARCH and GARCH Models: A Yule-Walker Approach

2020 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Financial Time Series ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Stationary Solutions ◽  
Finite Dimensional Subspace ◽  
Estimation Errors ◽  
Linear Processes ◽  
Finite Dimensional ◽  
Garch Processes ◽  
Asymptotic Upper Bound

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

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