Mathematics

We introduce and evaluate the degree of convexity of an unit ball, so-called, characteristic of convexity (COC) for the Grand Lebesgue Spaces, (GLS), which is a slight analog of the classical notion of the modulus of convexity (MOC).

Natural images are often the superposition of various parts of different geometric characteristics. For instance, an image might be a mixture of cartoon and texture structures. In addition, images are often given with missing data. In this paper, we develop a method for simultaneously decomposing an image to its two underlying parts and inpainting the missing data. Our separation inpainting method is based on and l 1 minimization approach, using two dictionaries, each sparsifying one of the image parts but not the other. We introduce a comprehensive convergence analysis of our method, in a general setting, utilizing the concepts of joint concentration, clustered sparsity, and cluster coherence. As the main application of our theory, we consider the problem of separating and inpainting an image to a cartoon and texture parts.

In this paper, we give formulas that allow one to move between transfer function type realizations of multi-variate Schur, Herglotz and Pick functions, without adding additional singularities except perhaps poles coming from the conformal transformation itself. In the two-variable commutative case, we use a canonical de Branges-Rovnyak model theory to obtain concrete realizations that analytically continue through the boundary for inner functions which are rational in one of the variables (so-called quasi-rational functions). We then establish a positive solution to McCarthy's Champagne conjecture for local to global matrix monotonicity in the settings of both two-variable quasi-rational functions and d -variable perspective functions.

The motivation behind this paper is threefold. Firstly, to study, characterize and realize operator concavity along with its applications to operator monotonicity of free functions on operator domains that are not assumed to be matrix convex. Secondly, to use the obtained Schur complement based representation formulas to analytically extend operator means of probability measures and to emphasize their study through random variables. Thirdly, to obtain these results in a decent generality. That is, for domains in arbitrary tensor product spaces of the form A⊗B(E) , where A is a Banach space and B(E) denotes the bounded linear operators over a Hilbert space E . Our arguments also apply when A is merely a locally convex space.

In this article, we analyze the approximation properties of the new family of Durrmeyer type exponential sampling operators. We derive the point-wise and uniform approximation theorem and Voronovskaya type theorem for these generalized family of operators. Further, we construct a convex type linear combination of these operators and establish the better approximation results. Finally, we provide few examples of the kernel functions to which the presented theory can be applied along with the graphical representation.

We analyse the behaviour of the exponential sampling series S ? w f at jump discontinuity of the bounded signal f. We obtain a representation lemma that is used for analysing the series S ? w f and we establish approximation of jump discontinuity functions by the series S ? w f. The rate of approximation of the exponential sampling series S ? w f is obtained in terms of logarithmic modulus of continuity of functions and the round-off and time-jitter errors are also studied. Finally we give some graphical representation of approximation of discontinuous functions by S ? w f using suitable kernels.

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces B α q ( L p ) in arbitrary dimension d , on bounded or unbounded domains. We show that RePU networks with a fixed activation function attain optimal approximation rates for functions in the Besov space B α τ ( L τ ) on the critical embedding line 1/τ=α/d+1/p for arbitrary smoothness order α>0 . Moreover, we show that ReLU networks attain near to optimal rates for any Besov space strictly above the critical line. Using interpolation theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.

The Kantorovich exponential sampling series at jump discontinuities of the bounded measurable signal f has been analysed. A representation lemma for the series is established and using this lemma certain approximation theorems for discontinuous signals are proved. The degree of approximation in terms of logarithmic modulus of smoothness for the series is studied. Further a linear prediction of signals based on past sample values has been obtained. Some numerical simulations are performed to validate the approximation of discontinuous signals f by the sampling series.

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

Given a Sobolev homeomorphism f∈ W 2,1 in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of ϵ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the W 2,1 norm on this set.