On approximation of hypersingular integral operators by bounded ones

Бабенко, В. Ф.; Коваленко, О. В.; Parfinovych, N. V.

doi:10.1016/j.jmaa.2022.126215

Cited by 6 publications

(2 citation statements)

References 20 publications

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“…). Now we state the well-known Gagliardo-Nirenberg embedding theorem (see [36, Lecture II]) that will plays an important role in our further analysis (for more details and other types of embedding, we refer to [9][10][11]).…”

Section: Functional Spacesmentioning

confidence: 99%

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

Kogut

Kohut

Parfinovych

2022

JODEA

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This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem.

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Section: Functional Spacesmentioning

confidence: 99%

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

Kogut

Kohut

Parfinovych

2022

JODEA

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“…In the case, when Q is a convex bounded domain, such problem was solved in [2]. Solutions to some extremal problems for similar classes of functions can be found in [8,9,11,6,4,3].…”

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confidence: 99%

On optimization of cubature formulae for Sobolev classes of functions defined on star domains

Kovalenko

2024

Mat. Stud.

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We find asymptotically optimal methods of recovery of the integration operator given values of the function at a finite number of points for a class of multivariate functions defined on a bounded star domain that have bounded in $L_p$ norm of their distributional gradient. Thus we generalize the known solution of this optimization problem in the case, when the domain of the functions is convex. Let $Q\subset \mathbb{R}^d$, $d\in\mathbb{N}$, be a nonempty bounded open set. By $W^{1,p}(Q)$, $p\in [1,\infty]$, we denote the Sobolev space of functions $f\colon Q\to \mathbb{R}$ such that $f$ and all their (distributional) partial derivatives of the first order belong to $L_p(Q)$. For $x=(x^1,\dots, x^d)\in \mathbb{R}^d$ and $q\in [1,\infty)$ we set$|x|_q:= \Big(\sum_{k=1}^d|x^k|^q\Big)^\frac {1}{q},$ $|x|_\infty:= \max\{|x^k|\colon k\in\{1,\ldots, d\}\}$, and $W^{\infty}_{p}(Q):=\{f\in W^{1,p}(Q)\colon \|\,|\nabla f|_1\,\|_{L_p(Q)}\leq 1\},$ where $\nabla f=(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_d})$, $p\in[1,\infty]$. In particular we prove the following statement: Let $d\geq 2$, $p\in(d,\infty]$ and $Q$ be a bounded star domain. Then$\displaystyle E_n\Big(W_{p}^{\infty}(Q)\Big)=c(d,p)\Big(\frac {\mathop{mes} Q}{2^d}\Big)^{\frac 1 d +\frac 1 {p'}}\cdot \frac{1+o(1)} {n^{\frac 1 d}}$ $(n\to\infty),$ where $E_n(X):=\inf\Big\{\inf\big\{ e(X,\Phi,x_1,\dots,x_n)\colon\, \Phi\colon\mathbb{R}^n\to\mathbb{R}\big\}\colon x_1,\dots,x_n\in Q\big\},$$e(X, \Phi, x_1,\dots,x_n):= \sup\Big\{\Big|\,\int\limits_{Q}f(x)dx - \Phi(f(x_1),\ldots,f(x_n))\Big|\colon f\in X\Big\}$for $X=W_{p}^{\infty}(Q)$, and $c(d,p)\in \mathbb{R}$ depends only on $d$ and $p$.

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On a general approach to some problems of approximation of operators

Kovalenko

2024

J Math Sci

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On approximation of hypersingular integral operators by bounded ones

Cited by 6 publications

References 20 publications

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

On optimization of cubature formulae for Sobolev classes of functions defined on star domains

On a general approach to some problems of approximation of operators

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