Class Of Function Spaces Research Articles

Moduli of smoothness, K-functionals and Jackson-type inequalities associated with kernel function approximation in learning theory

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a [Formula: see text]-functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness.

Hausdorff operators on some classical spaces of analytic functions

Abstract In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators $$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$ on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $ -measurable function on the open unit disk $\mathbb {D}$ , and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$ .

Subharmonic additions to the Beurling–Malliavin Theorems. II. On the completeness radius

The Beurling–Malliavin Theorem on the multiplier, considered in a subharmonic framework in the first part of our work, already in its original classical version within the framework of entire functions of exponential type, allowed in the 1960s to completely solve the problem of the radius of completeness of exponential system in the form of a remarkable criterion and exclusively in geometric terms for the exponents of this exponential system without any additional restrictions on the relative position of these exponents. The exact formulations of the Beurling– Malliavin Theorem on the radius of completeness in the introduction are somewhat adapted as a problem of the possible minimum growth along the real axis R of subharmonic functions with given constraints on their Riesz distributions of masses. In this largely overview part of the paper, we discuss the Beurling–Malliavin Theorem on the radius of completeness, along with its somewhat more general subharmonic manifestations. Thus, our results from 2014-16 allow us to obtain significantly more subtle results with respect to the Beurling – Malliavin Theorem itself on the radius of completeness with a defect excess of no more than 1 or 2 for exponents in classical rigid Banach spaces of continuous functions on a segment of fixed length or functions with integrable module in the p-th degree on such segments.

Generalized grand Lorentz martingale spaces

In this article, we define a new class of function spaces L_{{p,q}),\theta} which unify and generalize the Iwaniec–Sbordone spaces with q=p , grand Lorentz spaces \theta=1 and classical Lorentz spaces \theta=0 . Based on the new space, we introduce a kind of Hardy-type space, H^s_{{p,q}),\theta} , via the martingale operators and then develop a theory of these martingale Hardy spaces. More specifically, the atomic decompositions of H^s_{{p,q}),\theta} (0<p\leq1 , 1<q<\infty) are established. As applications, the dual theorems for the new framework are provided. We also obtain a new John–Nirenberg-type inequality by the duality. The results extend the very recent work of Jiao et al. (2017) to the case of generalized grand Lorentz spaces. Our results are new, even for the grand Lorentz spaces.

Sobolev-Type Embeddings for Neural Network Approximation Spaces

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in L^p) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of p. Furthermore, we derive sharp embeddings of these approximation spaces into Hölder spaces. We find that, analogous to the case of classical function spaces (such as Sobolev spaces, or Besov spaces) it is possible to trade “smoothness” (i.e., approximation rate) for increased integrability. Combined with our earlier results in Grohs and Voigtlaender (Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces, 2021. arXiv preprint arXiv:2104.02746), our embedding theorems imply a somewhat surprising fact related to “learning” functions from a given neural network space based on point samples: if accuracy is measured with respect to the uniform norm, then an optimal “learning” algorithm for reconstructing functions that are well approximable by ReLU neural networks is simply given by piecewise constant interpolation on a tensor product grid.

Boundedness of the discrete Hilbert transform in discrete Hölder spaces

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory. The Hilbert transform was the motivation for the development of modern harmonic analysis. Its discrete version is also widely used in many areas of science and technology and plays an important role in digital signal processing. The essential motivation behind thinking about discrete transforms is that experimental data are most frequently not taken in a continuous manner but sampled at discrete time values. Since much of the data collected in both the physical sciences and engineering are discrete, the discrete Hilbert transform is a rather useful tool in these areas for the general analysis of this type of data. The Hilbert transform has been well studied on classical function spaces such as H¨older, Lebesgue, Morrey, etc. But its discrete version, which also has numerous applications, has not been fully studied in discrete analogues of these spaces. In this paper, we discuss the discrete Hilbert transform in discrete H¨older spaces and obtain its boundedness in these spaces.

Projective descriptions of spaces of functions and distributions

AbstractWe present projective descriptions of classical spaces of functions and distributions. More precisely, we provide descriptions of these spaces by seminorms, which are defined by a combination of classical norms and multiplication or convolution with certain functions. These seminorms are simpler than the ones given by a supremum over bounded or compact sets.

Estimates for evolutionary partial differential equations in classical function spaces

Abstract We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

Two stars theorems for traces of the Zygmund space

For a Banach space X X defined in terms of a big- O O condition and its subspace x defined by the corresponding little- o o condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of x is naturally isometrically isomorphic to X X . The property is known for pairs of many classical function spaces (such as ( ℓ ∞ , c 0 ) (\ell _\infty , c_0) , (BMO, VMO), (Lip, lip), etc.) and plays an important role in the study of their geometric structure. The present paper is devoted to the biduality property for traces to closed subsets S ⊂ R n S\subset \mathbb {R}^n of a generalized Zygmund space Z ω ( R n ) Z^\omega (\mathbb {R}^n) . The method of the proof is based on a careful analysis of the structure of geometric preduals of the trace spaces along with a powerful finiteness theorem for the trace spaces Z ω ( R n ) | S Z^\omega (\mathbb {R}^n)|_S .

Translation and modulation invariant Banach spaces of tempered distributions satisfy the metric approximation property

In this paper, we establish the validity of the so-called Bounded Approximation Property (BAP) for a comprehensive class of translation and modulation invariant Banach spaces (B, ∥ · ∥B) of tempered distributions on the Euclidean space [Formula: see text]. In fact, such spaces have a double module structure, over some Beurling algebra with respect to convolution, and with respect to pointwise multiplication over some Fourier Beurling algebra. Combining this double module structure with functional analytic arguments which describe the approximation of convolution operators by discrete convolutions we are able to verify the BAP, in fact, for most cases even the Metric Approximation Property (MAP) for such Banach space. The family of spaces under consideration is very rich and contains virtually all the classical function spaces relevant for mathematical analysis, as long as the Schwartz space [Formula: see text] is dense in (B, ∥ · ∥B). In particular, all the reflexive spaces in this family are included. Moreover, this family of Banach spaces is closed with respect to intersections, sums and various interpolation methods.

Extension of Q‐type spaces related with weight functions via convolution operators on Heisenberg groups

In this paper, in the setting of Heisenberg groups , we introduce two classes of ‐type spaces related with weight functions denoted by . We investigate several basic properties of , including the average oscillation property, the John–Nirenberg‐type inequality and the relations with classical function spaces. By the family of convolution operators , we extend , to the function spaces on the Siegel upper half space . The wavelet characterizations of are also obtained.

Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces

We compute the exact value of the essential norm of a generalized Hilbert matrix operator acting on weighted Bergman spaces Avp and weighted Banach spaces Hv∞ of analytic functions, where v is a general radial weight. In particular, we obtain the exact value of the essential norm of the classical Hilbert matrix operator on standard weighted Bergman spaces Aαp for p>2+α,α≥0, and on Korenblum spaces Hα∞ for 0<α<1. We also cover the Hardy space Hp,1<p<∞, case. In the weighted Bergman space case, the essential norm of the Hilbert matrix is equal to the conjectured value of its operator norm and similarly in the Hardy space case the essential norm and the operator norm coincide. We also compute the exact value of the norm of the Hilbert matrix on Hwα∞ with weights wα(z)=(1−|z|)α for all 0<α<1. Also in this case, the values of the norm and essential norm coincide.

A NOTE ON A NEW SHARP RESULT IN SPACES OF PLURIHARMONIC FUNCTIONS AND RELATED PROBLEMS

We provide a new sharp result for classical Bergman spaces of pluriharmonic functions related with the trace operator extending previously known assertions. Related new estimates for other pluriharmonic new function spaces in product domains will be also discussed and provided.

О ЧИСЛЕННЫХ МЕТОДАХ РЕШЕНИЯ БЕСКОНЕЧНЫХ СИСТЕМ ЛИНЕЙНЫХ АЛГЕБРАИЧЕСКИХ УРАВНЕНИЙ

We provide a new sharp result for classical Bergman spaces of pluriharmonic functions related with the trace operator extending previously known assertions. Related new estimates for other pluriharmonic new function spaces in product domains will be also discussed and provided.

Approximation in shift-invariant spaces with deep ReLU neural networks

We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks. The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained. We also give lower bounds of the Lp(1≤p≤∞) approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.

Lower bounds for integration and recovery in L2

Function values are, in some sense, “almost as good” as general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert space into L2 are square-summable. Here we mainly prove new lower bounds. In particular we prove that the sampling numbers behave worse than the approximation numbers for Sobolev spaces with small smoothness. Hence there can be a logarithmic gap also in the case where the singular numbers of the embedding are square-summable. We first prove new lower bounds for the integration problem, again for rather classical Sobolev spaces of periodic univariate functions.

Convergence of linear combinations of iterates of an inner function

Let f be an inner function with f(0)=0 which is not a rotation and let fn be its n-th iterate. Let {an} be a sequence of complex numbers. We prove that the series ∑anfn(ξ) converges at almost every point ξ of the unit circle if and only if ∑|an|2<∞. The main step in the proof is to show that under this assumption, the function F=∑anfn has bounded mean oscillation. We also prove that F is bounded on the unit disc if and only if ∑|an|<∞. Finally we describe the sequences of coefficients {an} such that F belongs to other classical function spaces, as the disc algebra and the Dirichlet class.

Composition operators on Herz-type Triebel–Lizorkin spaces with application to semilinear parabolic equations

Let $$G:\mathbb {R}\rightarrow \mathbb {R}$$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such that $$\begin{aligned} \{G(f):f\in {\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset {\dot{K}} _{p,q}^{\alpha }F_{\beta }^{s} \end{aligned}$$ holds. Here $${\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}$$ are Herz-type Triebel–Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel–Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabolic equations $$\begin{aligned} \partial _{t}u-\Delta u=G(u) \end{aligned}$$ with initial data in Herz-type Triebel–Lizorkin spaces. Our results cover the results obtained with initial data in some known function spaces such us fractional Sobolev spaces. Some limit cases are given.

New classes of function spaces and singular operators

This article is dedicated to the memory of Garnik Al’bertovich Karapetyan and it contains a review of results obtained by G. A. Karapetyan and his colleagues within the joint Russian–Armenian project of RFBR. In the first section, we look at multi-anisotropic spaces which were intensively studied by Karapetyan and his students. The second section is devoted to a new class of singular Hausdorff and Hausdorff–Berezin operators. In the third section, we study the connection between real function spaces and operator algebras in a Hilbert space, established by means of a quantization procedure. UDK: 517.518.

Semilinear parabolic equations in Herz spaces

In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces.