I. Kh. Musin
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result for description of strong dual for weighted spaces of infinitely differentiable functions on real line and weighted spaces of sequences of infinitely differentiable functions on real line in terms of the Fourier-Laplace transform of functionals.
We focus on the study of $p$-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the genus properties in critical point theory, we establish some new criteria to guarantee the existence of infinitely many weak solutions for the considered problem.
We present a result of existence of infinitely many solutions for the Dirichlet problem involving the p-Laplacian in annular domains, when $p\leq N$, contouring the failure of compactness of $W^{1,p}(Ω)$ in $C^0(\barΩ)$ applying a variable change.
In this paper we establish exact order estimates for the best uniform orthogonal trigonometric approximations of the classes of $2π$-periodic functions, whose $(ψ,β)$-derivatives belong to unit balls of spaces $L_{p}$, $1\leq p<\infty$, in the case, when the sequence $ψ(k)$ tends to zero faster, than any power function, but slower than geometric progression. Similar estimates are also established in the $L_{s}$-metric, $1<s\leq\infty$ for the classes of differentiable functions, which $(ψ,β)$-derivatives belong to unit ball of space $L_{1}$.
We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2π$-periodic continuous functions, which $(ψ,β)$-derivatives $f_β^ψ$ belong to unit balls of spaces $L_{p}, 1\leq p<\infty$ in case at consequences $ψ(k)$ decrease to nought faster than any power function. We also established the analogical estimations in $L_{s}$-metric, $1<s\leq\infty$, for classes of the summable $(ψ,β)$-differentiable functions, such that $\parallel f_β^ψ\parallel_{1}\leq1$.
We obtain asymptotic estimates for the best approximations by trigonometric polynomials in the metric space $C$ $(L_p)$ of classes of periodic functions that can be represented as a convolution of kernels $Ψ_β$, which Fourier coefficients tend to zero faster than any power sequence, with functions $φ\in C (φ\in L_p),$ which moduli of continuity do not exceed a fixed majorant $ω(t)$. It is proved that in the spaces $C$ and $L_1$ the obtained estimates are asymptotically exact for convex moduli of continuity $ω(t)$.
Quasianalytic classes are classes of infinitely differentiable functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of infinitely differentiable functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of infinitely differentiable by real-analytic functions.
We find uniform with respect to parameter $p \ (1\leq p\leq\infty)$ upper estimations of best approximations by trigonometric polynomials of classes $C^ψ_{β,p}$ of periodic functions generated by sequences $ψ(k)$, that decrease to nought faster than any power function. Obtained estimations are exact for order and have constants that are written in explicit form and depend on function $ψ$ only. We obtain analogical estimations for best approximations of classes $L^ψ_{β,1}$ in metrics of spaces $L_{s}$, $1\leq s\leq \infty$.
This is the English translation of my old paper 'Definición y estudio de una función indefinidamente diferenciable de soporte compacto', Rev. Real Acad. Ciencias 76 (1982) 21-38. In it a function (essentially Fabius function) is defined and given its main properties, including: unicity, interpretation as a probability, partition of unity with its translates, formulas for its $n$-th derivates, rationality of its values at dyadic points, formulas for the effective computation of these values, and some arithmetical properties of these values. Since I need it now for a reference, I have translated…
Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$ with elastic impact on the boundary of $K$ is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulation it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is…
The Hobby-Rice Theorem states that, given $n$ functions $f_j$ on ${\mathbb{R}}^N$, there exists a multiplier $h$ such that the integrals of $f_jh$ are all simultaneously zero. This multiplier takes values~$\pm1$ and is discontinuous. We show how to find a multiplier $h=e^{ig}$ that is infinitely differentiable, takes values on the unit circle, and is such that the integrals of $f_jh$ are all zero. We also show the existence of $n$ infinitely differentiable, real functions $g_j$ such that the $n$ functions $f_j e^{ig_j}$ are pairwise orthogonal.
A glimpse at $\sum{{\sin(kx)}\over{k}}$ gives a few lines exposition of Fourier series's quadratic mean convergence for square integrable functions and Dirichlet's convergence theorem of the Fourier serie of a piecewise differentiable function with integrable derivative.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。