# Numerical Functional Analysis and Optimization

Published by Taylor & Francis

Online ISSN: 1532-2467

Print ISSN: 0163-0563

Published by Taylor & Francis

Online ISSN: 1532-2467

Print ISSN: 0163-0563

Publications

The spectral analysis of discretized one-dimensional Schr\"{o}dinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is that of finding finite dimensional matrices whose eigenvalues approximate the spectrum of an infinite dimensional operator. In this note we observe that the seminal work of Pimsner-Voiculescu on AF embeddings of irrational rotation algebras provides a nice answer to the finite dimensional spectral approximation problem for a broad class of operators including the quasiperiodic case of the Schr\"{o}dinger operators mentioned above. Indeed, the theory of continued fractions not only provides good matrix models for spectral computations (i.e. the Pimsner-Voiculescu construction) but also yields {\em sharp} rates of convergence for spectral approximations of operators in irrational rotation algebras.

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- 28 Reads
- 16 Citations

We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation.

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- 46 Reads
- 23 Citations

Let $X$ be a reflexive Banach space. In this paper we give a necessary and sufficient condition for an operator $T\in \mathcal{K}(X)$ to have the best approximation in numerical radius from the convex subset $\mathcal{U} \subset \mathcal{K}(X),$ where $\mathcal{K}(X)$ denotes the set of all linear, compact operators from $X$ into $X.$ We will also present an application to minimal extensions with respect to the numerical radius. In particular some results on best approximation in norm will be generalized to the case of the numerical radius. Comment: 13 pages

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- 113 Reads
- 9 Citations

Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.

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- 134 Reads
- 17 Citations

We consider the minimization of the number of non-zero coefficients (the
$\ell_0$ "norm") of the representation of a data set in terms of a dictionary
under a fidelity constraint. (Both the dictionary and the norm defining the
constraint are arbitrary.) This (nonconvex) optimization problem naturally
leads to the sparsest representations, compared with other functionals instead
of the $\ell_0$ "norm". Our goal is to measure the sets of data yielding a
$K$-sparse solution--i.e. involving $K$ non-zero components. Data are assumed
uniformly distributed on a domain defined by any norm--to be chosen by the
user. A precise description of these sets of data is given and relevant bounds
on the Lebesgue measure of these sets are derived. They naturally lead to bound
the probability of getting a $K$-sparse solution. We also express the
expectation of the number of non-zero components. We further specify these
results in the case of the Euclidean norm, the dictionary being arbitrary.

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- 65 Reads
- 3 Citations

Well-posed models and computational algorithms are developed and analyzed for control of a class of partial differential equations that describe the motions of thermo-viscoelastic structures. An abstract (state space) framework and a general well-posedness result are presented that can be applied to a large class of thermo-elastic and thermo-viscoelastic models. This state space framework is used in the development of a computational scheme to be used in the solution of a linear quadratic regulator (LQR) control problem. A detailed convergence proof is provided for the viscoelastic model and several numerical results are presented to illustrate the theory and to analyze problems for which the theory is incomplete.

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- 41 Reads
- 12 Citations

Formulae for the value of a harmonic function at the center of a rectangle
are found that involve boundary integrals. The central value of a harmonic
function is shown to be well approximated by the mean value of the function on
the boundary plus a very small number (often just 1 or 2) of additional
boundary integrals. The formulae are consequences of Steklov (spectral)
representations of the functions that converge exponentially at the center.
Similar approximation are found for the central values of solutions of Robin
and Neumann boundary value problems. The results are based on explicit
expressions for the Steklov eigenvalues and eigenfunctions.

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- 115 Reads
- 10 Citations

For some fractal measures it is a very difficult problem in general to prove
the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In
fact there are examples of extremely sparse sets that are not even Bessel
spectra. In this paper we investigate this problem for general fractal measures
induced by iterated function systems (IFS). We prove some existence results of
spectra associated with Hadamard pairs. We also obtain some characterizations
of Bessel spectrum in terms of finite matrices for affine IFS measures, and one
sufficient condition of frame spectrum in the case that the affine IFS has no
overlap.

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- 45 Reads

Recently, Martin Hutzenthaler pointed out that the explicit Euler method
fails to converge strongly to the exact solution of a stochastic differential
equation (SDE) with superlinearly growing and globally one sided Lipschitz
drift coefficient. Afterwards, he proposed an explicit and easily implementable
Euler method, i.e tamed Euler method, for such an SDE and showed that this
method converges strongly with order of one half. In this paper, we use the
tamed Euler method to solve the stochastic differential equations with
piecewise continuous arguments (SEPCAs) with superlinearly growing coefficients
and prove that this method is convergent with strong order one half.

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- 118 Reads
- 8 Citations

We consider the diffusion equation in the setting of operator theory. In particular, we study the characterization of the limit of the diffusion operator for diffusivities approaching zero on a subdomain $\Omega_1$ of the domain of integration of $\Omega$. We generalize Lions' results to covering the case of diffusivities which are piecewise $C^1$ up to the boundary of $\Omega_1$ and $\Omega_2$, where $\Omega_2 := \Omega \setminus \overline{\Omega}_1$ instead of piecewise constant coefficients. In addition, we extend both Lions' and our previous results by providing the strong convergence of $(A_{\bar{p}_\nu}^{-1})_{\nu \in \mathbb{N}^\ast},$ for a monotonically decreasing sequence of diffusivities $(\bar{p}_\nu )_{\nu \in \mathbb{N}^\ast}$.

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- 31 Reads
- 6 Citations

In this paper we apply methods of proof mining to obtain a uniform effective
rate of asymptotic regularity for the Mann iteration associated to
$\kappa$-strict pseudo-contractions on convex subsets of Hilbert spaces.

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- 35 Reads
- 7 Citations

Function spaces are central topic in analysis. Often those spaces and related
analysis involves symmetries in form of an action of a Lie group. Coorbit
theory as introduced by Feichtinger and Gr\"ochenig and then later extended in
[3] gives a unified method to construct Banach spaces of functions based on
representations of Lie groups. In this article we identify the homogeneous
Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in
the sense of [3] and use this to derive atomic decompositions for the Besov
spaces.

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- 123 Reads
- 14 Citations

Kuhn-Tucker conditions for mathematical programming problems in Banach spaces
partially ordered by cone with empty interior are obtained under strong
simultaneity condition. If partial ordered cone has interior point, it is
proved that Slater and strong simultaneity conditions are equivalent.

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- 45 Reads
- 2 Citations

In this paper, we extend the Moreau (Riesz) decomposition theorem, from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dimX⩾3(dimX⩽2) is strongly orthogonally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex Banach space), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.

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- 73 Reads
- 8 Citations

A recurrent theme in functional analysis is the interplay between the theory
of positive definite functions, and their reproducing kernels, on the one hand,
and Gaussian stochastic processes, on the other. This central theme is
motivated by a host of applications, e.g., in mathematical physics, and in
stochastic differential equations, and their use in financial models. In this
paper, we show that, for three classes of cases in the correspondence, it is
possible to obtain explicit formulas which are amenable to computations of the
respective Gaussian stochastic processes. For achieving this, we first develop
two functional analytic tools. They are: $(i)$ an identification of a universal
sample space $\Omega$ where we may realize the particular Gaussian processes in
the correspondence; and (ii) a procedure for discretizing computations in
$\Omega$. The three classes of processes we study are as follows: Processes
associated with: (a) arbitrarily given sigma finite regular measures on a fixed
Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with
systems of self-similar measures arising in the theory of iterated function
systems. Even our results in (a) go beyond what has been obtained previously,
in that earlier studies have focused on more narrow classes of measures,
typically Borel measures on $\mathbb R^n$. In our last theorem (section 10),
starting with a non-degenerate positive definite function $K$ on some fixed set
$T$, we show that there is a choice of a universal sample space $\Omega$, which
can be realized as a "boundary" of $(T, K)$. Its boundary-theoretic properties
are analyzed, and we point out their relevance to the study of electrical
networks on countable infinite graphs.

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- 85 Reads
- 34 Citations

We study the approximation of fixed points of nonexpansive mappings in CAT(k)
spaces. We show that the iterative sequence generated by the Moudafi's
viscosity type algorithm converges to one of the fixed points of the
nonexpansive mapping depending on the contraction applied in the algorithm.

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- 154 Reads
- 15 Citations

In this paper, error estimates are presented for a certain class of optimal
control problems with elliptic PDE-constraints. It is assumed that in the cost
functional the state is measured in terms of the energy norm generated by the
state equation. The functional a posteriori error estimates developed by Repin
in late 90's are applied to estimate the cost function value from both sides
without requiring the exact solution of the state equation. Moreover, a lower
bound for the minimal cost functional value is derived. A meaningful error
quantity coinciding with the gap between the cost functional values of an
arbitrary admissible control and the optimal control is introduced. This error
quantity can be estimated from both sides using the estimates for the cost
functional value. The theoretical results are confirmed by numerical tests.

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- 20 Reads
- 1 Citation

An equilateral dimension of a normed space is a maximal number of pairwise
equidistant points of this space. The aim of this paper is to study the
equilateral dimension of certain classes of finite dimensional normed spaces.
The well-known conjecture states that the equilateral dimension of any
$n$-dimensional normed space is not less than $n+1$. By using an elementary
continuity argument, we establish it in the following classes of spaces:
permutation-invariant spaces, Orlicz-Musielak spaces and in one codimensional
subspaces of $\ell^n_{\infty}$. For smooth and symmetric spaces,
Orlicz-Musielak spaces satisfying an additional condition and every
$(n-1)$-dimensional subspace of $\ell^{n}_{\infty}$ we also provide some weaker
bounds on the equilateral dimension for every space which is sufficiently close
to one of these. This generalizes the result of Swanepoel and Villa concerning
the $\ell_p^n$ spaces.

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- 42 Reads
- 6 Citations

In this work we combine Wiener chaos expansion approach to study the dynamics
of a stochastic system with the classical problem of the prediction of a
Gaussian process based on part of its sample path. This is done by considering
special bases for the Gaussian space $\mathcal G$ generated by the process,
which allows us to obtain an orthogonal basis for the Fock space of $\mathcal
G$ such that each basis element is either measurable or independent with
respect to the given samples. This allows us to easily derive the chaos
expansion of a random variable conditioned on part of the sample path. We
provide a general method for the construction of such basis when the underlying
process is Gaussian with stationary increment. We evaluate the bases elements
in the case of the fractional Brownian motion, which leads to a prediction
formula for this process.

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- 53 Reads
- 6 Citations

We introduce new types of systems of generalized quasi-variational
inequalities and we prove the existence of the solutions by using results of
pair equilibrium existence for free abstract economies. We consider the fuzzy
models and we also introduce the random free abstract economy and the random
equilibrium pair. The existence of the solutions for the systems of
quasi-variational inequalities comes as consequences of the existence of
equilibrium pairs for the considered free abstract economies.

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- 25 Reads

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t), u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity. Comment: 26 pages

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- 87 Reads
- 8 Citations

The Rayleigh conjecture on the representation of the scattered field in the exterior of an obstacle $D$ is widely used in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), and in this paper we present successful numerical algorithms based on the MRC for various 2D and 3D obstacle scattering problems. The 3D obstacles include a cube and an ellipsoid. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods.

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- 45 Reads
- 5 Citations

In this paper, a kind of non regular constraints and a principle for seeking
critical point under the constraint are presented, where no Lagrange multiplier
is involved. Let $E, F$ be two Banach spaces, $g: E\rightarrow F$ a $c^1$ map
defined on an open set $U$ in $E,$ and the constraint $S=$ the preimage
$g^{-1}(y_0), y_0\in F.$ A main deference between the non regular constraint
and regular constraint is that $g'(x)$ at any $x\in S$ is not surjective.
Recently, the critical point theory under the non regular constraint is a
concerned focus in optimization theory. The principle also suits the case of
regular constraint. Coordinately, the generalized regular constraint is
introduced, and the critical point principle on generalized regular constraint
is established. Let $f: U \rightarrow \mathbb{R}$ be a nonlinear functional.
While the Lagrange multiplier $L$ in classical critical point principle is
considered, and its expression is given by using generalized inverse
${g'}^+(x)$ of $g'(x)$ as follows : if $x\in S$ is a critical point of $f|_S,$
then $L=f'(x)\circ {g'}^+(x) \in F^*.$ Moreover, it is proved that if $S$ is a
regular constraint, then the Lagrange multiplier $L$ is unique; otherwise, $L$
is ill-posed. Hence, in case of the non regular constraint, it is very
difficult to solve Euler equations, however, it is often the case in
optimization theory. So the principle here seems to be new and applicable. By
the way, the following theorem is proved; if $A\in B(E,F)$ is double split,
then the set of all generalized inverses of $A,$ $GI(A)$ is smooth
diffeomorphic to certain Banach space. This is a new and interesting result in
generalized inverse analysis.

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- 20 Reads
- 3 Citations

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed, and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting (see Schuster et al. [1313.
T. Schuster ,
B. Kaltenbacher ,
B. Hofmann , and
K. S. Kazimierski ( 2012 ). Regularization Methods in Banach Spaces . Radon Series on Computational and Applied Mathematics, Vol. 10. Walter de Gruyter GmbH & Co. KG, Berlin. View all references]). We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer [1111.
A. Neubauer ( 1997 ). On converse and saturation results for Tikhonov regularization of linear ill-posed problems . SIAM J. Numer. Anal. 34 : 517 – 527 . View all references], and generalize existing convergence rates results. The key tool in our analysis is a homogeneous source condition, which we put into relation to the other existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces (see Groetsch [77.
Groetsch , C. W. ( 1984 ). The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind . Pitman , Boston . View all references]).

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- 114 Reads
- 19 Citations

Rates of convergence of solutions of various two-dimensional $\alpha-$regularization models, subject to periodic boundary conditions, toward solutions of the exact Navier-Stokes equations are given in the $L^\infty$-$L^2$ time-space norm, in terms of the regularization parameter $ \alpha$, when $\alpha$ approaches zero. Furthermore, as a paradigm, error estimates for the Galerkin approximation of the exact two-dimensional Leray-$\alpha$ model are also presented in the $L^\infty$-$L^2$ time-space norm. Simply by the triangle inequality, one can reach the error estimates of the solutions of Galerkin approximation of the $\alpha$-regularization models toward the exact solutions of the Navier-Stokes equations in the two-dimensional periodic boundary conditions case. Comment: 29 pages

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- 118 Reads
- 34 Citations

This work focus on convergence analysis of the projected gradient method for
solving constrained convex minimization problem in Hilbert spaces. We show that
the sequence of points generated by the method employing the Armijo linesearch
converges weakly to a solution of the considered convex optimization problem.
Weak convergence is established by assuming convexity and G\^ateaux
differentiability of the objective function, whose G\^ateaux derivative is
supposed to be uniformly continuous on bounded sets. Furthermore, we propose
some modifications in the classical projected gradient method in order to
obtain strong convergence. The new variant has the following desirable
properties: the sequence of generated points is entirely contained in a ball
with diameter equal to the distance between the initial point and the solution
set, and the whole sequence converges strongly to the solution of the problem
that lies closest to the initial iterate. Convergence analysis of both methods
is presented without Lipschitz continuity assumption.

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- 553 Reads
- 21 Citations

We obtain series expansion formulas for the Hadamard fractional integral and
fractional derivative of a smooth function. When considering finite sums only,
an upper bound for the error is given. Numerical simulations show the
efficiency of the approximation method.

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- 126 Reads
- 65 Citations

Based on functional analysis, we propose an algorithm for finite-norm
solutions of higher-order linear Fuchsian-type ordinary differential equations
(ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only
the four arithmetical operations on integers. This algorithm is based on a
band-diagonal matrix representation of the differential operator P(x,d/dx),
though it is quite different from the usual Galerkin methods. This
representation is made for the respective CONSs of the input Hilbert space H
and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables
the construction of a recursive algorithm for solving the ODE. However, a
solution of the simultaneous linear equations represented by this matrix does
not necessarily correspond to the true solution of ODE. We show that when this
solution is an l^2 sequence, it corresponds to the true solution of ODE. We
invent a method based on an integer-type algorithm for extracting only l^2
components. Further, the concrete choice of Hilbert spaces H and H' is also
given for our algorithm when p_m is a polynomial or a rational function with
rational coefficients. We check how our algorithm works based on several
numerical demonstrations related to special functions, where the results show
that the accuracy of our method is extremely high.

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- 37 Reads
- 6 Citations

This paper is devoted to the study of a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo -- Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as. Finally, we give numerical results Comment: 20 pages

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- 164 Reads
- 7 Citations

In this article, an abstract framework for the error analysis of
discontinuous Galerkin methods for control constrained optimal control problems
is developed. The analysis establishes the best approximation result from a
priori analysis point of view and delivers reliable and efficient a posteriori
error estimators. The results are applicable to a variety of problems just
under the minimal regularity possessed by the well-posed ness of the problem.
Subsequently, applications of $C^0$ interior penalty methods for a boundary
control problem as well as a distributed control problem governed by the
biharmonic equation subject to simply supported boundary conditions are
discussed through the abstract analysis. Numerical experiments illustrate the
theoretical findings. Finally, we also discuss the variational discontinuous
discretization method (without discretizing the control) and its corresponding
error estimates.

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- 118 Reads
- 16 Citations

We study the finite-dimensional spaces V, that are invariant under the action of the finite differences operator . Concretely, we prove that if V is such an space, there exists a finite-dimensional translation invariant space W such that V ⊆ W. In particular, all elements of V are exponential polynomials. Furthermore, V admits a decomposition V = P ⊕ E with P a space of polynomials and E a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by Montel [77.
M. P. Montel ( 1937 ). Sur quelques extensions d'un théorème de Jacobi . Prace Matematyczno-Fizyczne 44 ( 1 ): 315 – 329 . View all references], which states that, if f: ℝ → ℂ is a continuous function satisfying for all t ∈ ℝ and certain h 1, h 2 ∈ ℝ∖{0} such that h 1/h 2 ∉ ℚ, then f(t) = a 0 + a 1t + … +a m−1t m−1 for all t ∈ ℝ and certain complex numbers a 0, a 1,…, a m−1. We demonstrate, with quite different arguments, the same result not only for ordinary functions f(t) but also for complex valued distributions. Finally, we also consider the subspaces V that are Δh 1h 2…h m -invariant for all h 1,…, h m ∈ ℝ.

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- 55 Reads
- 18 Citations

Weak sharp minimality is a notion emerged in optimization, whose utility is
largeley recognized in the convergence analysis of algorithms for solving
extremum problems as well as in the study of the perturbation behaviour of such
problems. In the present paper some dual constructions of nonsmooth analysis,
mainly related to quasidifferential calculus and its recent developments, are
employed in formulating sufficient conditions for global weak sharp minimality.
They extend to nonconvex functions a condition, which is known to be valid in
the convex case. A feature distinguishing the results here proposed is that
they avoid to assume the Asplund property on the underlying space.

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- 36 Reads
- 1 Citation

In this paper, we give sufficient
conditions for the existence of solutions of a general model which
includes as special cases many generalized vector quasi-equilibrium
problems with set-valued maps. The obtained results generalize and
improve several earlier results.

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- 36 Reads
- 2 Citations

We incorporate inertial terms in the hybrid proximal-extragradient algorithm
and investigate the convergence properties of the resulting iterative scheme
designed for finding the zeros of a maximally monotone operator in real Hilbert
spaces. The convergence analysis relies on extended Fej\'er monotonicity
techniques combined with the celebrated Opial Lemma. We also show that the
classical hybrid proximal-extragradient algorithm and the inertial versions of
the proximal point, the forward-backward and the forward-backward-forward
algorithms can be embedded in the framework of the proposed iterative scheme.

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- 104 Reads
- 72 Citations

Certain Bernoulli convolution measures (\mu) are known to be spectral.
Recently, much work has concentrated on determining conditions under which
orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure
known to be spectral, the orthonormal basis need not be unique; indeed, there
are often families of such spectral bases.
Let \lambda = 1/(2n) for a natural number n and consider the Bernoulli
measure (\mu) with scale factor \lambda. It is known that L^2(\mu) has a
Fourier basis. We first show that there are Cuntz operators acting on this
Hilbert space which create an orthogonal decomposition, thereby offering
powerful algorithms for computations for Fourier expansions.
When L^2(\mu) has more than one Fourier basis, there are natural unitary
operators U, indexed by a subset of odd scaling factors p; each U is defined by
mapping one ONB to another. We show that the unitary operator U can also be
orthogonally decomposed according to the Cuntz relations. Moreover, this
operator-fractal U exhibits its own self-similarity.

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- 47 Reads
- 13 Citations

The classical frame potential in a finite dimensional Hilbert space has been
introduced by Benedetto and Fickus, who showed that all finite unit-norm tight
frames can be characterized as the minimizers of this energy functional. This
was the start point of a series of new results in frame theory, related to
finding tight frames with determined length. The frame potential has been
studied in the traditional setting as well as in the finite-dimensional fusion
frame context. In this work we introduce the concept of {\sl mixed frame
potential}, which generalizes the notion of the Benedetto-Fickus frame
potential. We study properties of this new potential, and give the structure of
its critical pairs of sequences on a suitable restricted domain. For a given
sequence ${\alpha_m}_{m=1,...,N}$ in $K,$ where $K$ is $\mathbb{R}$ or
$\mathbb{C},$ we obtain necessary and sufficient conditions in order to have a
dual pair of frames ${f_m}_{m=1,...,N}$, ${g_m}_{m=1,...,N}$ such that $<f_m,
g_m>=\alpha_m$ for all $m=1,..., N.$

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- 42 Reads
- 2 Citations

We will show that tight frames satisfying the restricted isometry property
give rise to nearly tight fusion frames which are nearly orthogonal and hence
are nearly equi-isoclinic. We will also show how to replace parts of the RIP
frame with orthonormal sets while maintaining the RIP property.

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- 49 Reads
- 6 Citations

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower)
semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently,
for an upper semi-frame, the frame operator is bounded, but has an unbounded
inverse, whereas a lower semi-frame has an unbounded frame operator, with
bounded inverse. For upper semi-frames, in the discrete and the continuous
case, we build two natural Hilbert scales which may yield a novel
characterization of certain function spaces of interest in signal processing.
We present some examples and, in addition, some results concerning the duality
between lower and upper semi-frames, as well as some generalizations, including
fusion semi-frames and Banach semi-frames.

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- 338 Reads
- 25 Citations

Given a parametrized family of finite frames, we consider the optimization
problem of finding the member of this family whose coefficient space most
closely contains a given data vector. This nonlinear least squares problem
arises naturally in the context of a certain type of radar system. We derive
analytic expressions for the first and second partial derivatives of the
objective function in question, permitting this optimization problem to be
efficiently solved using Newton's method. We also consider how sensitive the
location of this minimizer is to noise in the data vector. We further provide
conditions under which one should expect the minimizer of this objective
function to be unique. We conclude by discussing a related
variational-calculus-based approach for solving this frame optimization problem
over an interval of time.

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- 26 Reads

A generalization with singular weights of Moore-Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.

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- 444 Reads
- 18 Citations

Given $A$ and $B$ two nonempty subsets in a metric space, a mapping $T : A
\cup B \rightarrow A \cup B$ is relatively nonexpansive if $d(Tx,Ty) \leq
d(x,y) \text{for every} x\in A, y\in B.$ A best proximity point for such a
mapping is a point $x \in A \cup B$ such that $d(x,Tx)=\text{dist}(A,B)$. In
this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P.
Veeramani, Proximal normal structure and relatively nonexpansive mappings,
Studia Math., 171 (2005), 283-293] for relatively nonexpansive mappings in
Banach spaces to more general metric spaces. Namely, we give existence results
of best proximity points for cyclic and noncyclic relatively nonexpansive
mappings in the context of Busemann convex reflexive metric spaces. Moreover,
particular results are proved in the setting of CAT(0) and uniformly convex
geodesic spaces. Finally, we show that proximal normal structure is a
sufficient but not necessary condition for the existence in $A \times B$ of a
pair of best proximity points.

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- 58 Reads
- 17 Citations

We consider the problem of reconstructing, from the interior data $u(x,1)$, a function $u$ satisfying a nonlinear elliptic equation $$ \Delta u = f(x,y,u(x,y)), x \in \RR, y > 0. $$ Comment: 24 pages

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- 43 Reads
- 1 Citation

In this paper, we consider localized integral operators whose kernels have
mild singularity near the diagonal and certain Holder regularity and decay off
the diagonal. Our model example is the Bessel potential operator ${\mathcal
J}_\gamma, \gamma>0$. We show that if such a localized integral operator has
stability on a weighted function space $L^p_w$ for some $p\in [1, \infty)$ and
Muckenhoupt $A_p$-weight $w$, then it has stability on weighted function spaces
$L^{p'}_{w'}$ for all $1\le p'<\infty$ and Muckenhoupt $A_{p'}$-weights $w'$.

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- 62 Reads
- 14 Citations

We set up a multiresolution analysis on fractal sets derived from limit sets
of Markov Interval Maps. For this we consider the $\mathbb{Z}$-convolution of a
non-atomic measure supported on the limit set of such systems and give a
thorough investigation of the space of square integrable functions with respect
to this measure. We define an abstract multiresolution analysis, prove the
existence of mother wavelets, and then apply these abstract results to Markov
Interval Maps. Even though, in our setting the corresponding scaling operators
are in general not unitary we are able to give a complete description of the
multiresolution analysis in terms of multiwavelets.

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- 38 Reads

The problem of the minimization of least squares functionals with $\ell^1$ penalties is considered in an infinite dimensional Hilbert space setting. While there are several algorithms available in the finite dimensional setting there are only a few of them which come with a proper convergence analysis in the infinite dimensional setting. In this work we provide an algorithm from a class which have not been considered for $\ell^1$ minimization before, namely a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy to implement algorithm. We present experiments which indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of $\ell^1$-minimization.

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- 50 Reads

We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level algorithms, based on the conjugate gradient method and the Landweber--Richardson method are proposed that determine the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of band-limited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms.

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- 13 Reads
- 16 Citations

An adaptive regularization strategy for stabilizing Newton-like iterations on
a coarse mesh is developed in the context of adaptive finite element methods
for nonlinear PDE. Existence, uniqueness and approximation properties are known
for finite element solutions of quasilinear problems assuming the initial mesh
is fine enough. Here, an adaptive method is started on a coarse mesh where the
finite element discretization and quadrature error produce a sequence of
approximate problems with indefinite and ill-conditioned Jacobians. The methods
of Tikhonov regularization and pseudo-transient continuation are related and
used to define a regularized iteration using a positive semidefinite penalty
term. The regularization matrix is adapted with the mesh refinements and its
scaling is adapted with the iterations to find an approximate sequence of
coarse mesh solutions leading to an efficient approximation of the PDE
solution. Local q-linear convergence is shown for the error and the residual in
the asymptotic regime and numerical examples of a model problem illustrate
distinct phases of the solution process and support the convergence theory.

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- 24 Reads
- 6 Citations

In this paper we study a general family of multivariable Gaussian stochastic
processes. Each process is prescribed by a fixed Borel measure $\sigma$ on
$\mathbb R^n$. The case when $\sigma$ is assumed absolutely continuous with
respect to Lebesgue measure was studied earlier in the literature, when $n=1$.
Our focus here is on showing how different equivalence classes (defined from
relative absolute continuity for pairs of measures) translate into concrete
spectral decompositions of the corresponding stochastic processes under study.
The measures $\sigma$ we consider are typically purely singular. Our proofs
rely on the theory of (singular) unbounded operators in Hilbert space, and
their spectral theory.

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- 48 Reads
- 44 Citations

This paper addresses Tikhonov like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates which combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results. Comment: 22 pages, submitted to "Numerical Functional Analysis and Optimization"

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- 96 Reads
- 34 Citations

The theorem of Gopengauz guarantees the existence of a polynomial which well approximates a function f 2 C q [Gamma1; 1], while at the same time its kth derivative (k q) well approximates the kth derivative of the function, and moreover the polynomial and its derivatives respectively interpolate the function and its derivatives at Sigma1. With more generality, we shall prescribe that the polynomial interpolate the function at up to q + 1 points near 1 and up to q + 1 points near Gamma1. The points may coalesce, in which case one also interpolates at the coalescent point a number of derivatives one less than the multiplicity of coalescence. Aside from intrinsic theoretical interest, our results are clearly applicable in describing more precisely the error incurred in certain linear processes of simultaneous approximation, such as interpolation with added nodes near Sigma1. The original theorem of Gopengauz will be shown to follow as a special case. AMS Subject Classification: Pr...

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- 49 Reads
- 5 Citations

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