arXiv subjects
Publications and source records attributed to Ivan Gadjev.
We study the behaviour of the smallest possible constant $d_n$ in weighted Hardy inequality $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2 k^\epsilon\le d(n,\epsilon)\,\sum_{k=1}^{n}{a_k^2}\,k^\epsilon $$ The exact rate of convergence of $d_n$ is established and the ``almost extremal'' sequence is found.
We study the behaviour of the smallest possible constant $d(a,b, p,\epsilon)$ in Hardy inequality $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^px^{\epsilon}\,dx\leq d(a,b,p,\epsilon)\,\int_a^b [f(x)]^px^{\epsilon}\, dx, \quad 2\le p<\infty. $$ The exact rate of convergence of $d(a,b,p,\epsilon)$ is established and the ``almost extremal'' function is found.
A new Goodman-Sharma type modification of the Meyer-K\"{o}nig and Zeller operator for approximation of bounded continuous functions on [0,1) is presented. We estimate the approximation error of the proposed operator and prove direct and strong converse theorems with respect to a related K-functional. The operator is linear but not a positive one. However it benefits a better order of approximation compared to the Goodman-Sharma variant of Meyer-K\"{o}nig and Zeller type operator investigated by Ivanov and Parvanov in 2012.
A new Goodman-Sharma modification of the Baskakov operator is presented for approximation of bounded and continuous on $[0,\,\infty)$ functions. In our study on the approximation error of the proposed operator we prove direct and strong converse theorems with respect to a related K-functional. This operator is linear but not positive. However it has the advantage of a higher order of approximation compared to the Goodman-Sharma variant of the Baskakov operator defined in 2005 by Finta.
Here we study the approximation properties of a modified Goodman-Sharma operator recently considered by Acu and Agrawal in 2019. This operator is linear but not positive. It has the advantage of a higher order of approximation of functions compared with the Goodman-Sharma operator. We prove direct and strong converse theorems in terms of a related K-functional.
We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^p\,dx\leq d(a,b)\,\int_a^b [f(x)]^p dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^p\leq d_n\,\sum_{k=1}^{n}a_k^p. $$ The exact rate of convergence of $d(a,b)$ and $d_n$ is established and the ``almost extremal'' function and sequence are found.
We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2. $$ The exact constant $d(a,b)$ and the precise rate of convergence of $d_n$ are established and the extremal function and the ``almost extremal'' sequence are found.
We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy's inequalities $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2, \qquad (a_1,\ldots,a_n) \in \mathbb{R}^n $$ and $$ \int_{0}^{\infty}\Bigg(\frac{1}{x}\int\limits_{0}^{x}f(t)\,dt\Bigg)^2 dx \leq c_n \int_{0}^{\infty} f^2(x)\,dx, \ \ f\in \mathcal{H}_n, $$ for the finite dimensional spaces $\mathbb{R}^n$ and $\mathcal{H}_n:=\{f\,:\, \int_0^x f(t) dt =e^{-x/2}\,p(x)\ :\ p\in \mathcal{P}_n, p(0)=0\}$, where $\mathcal{P}_n$ is the set of real-valued algebraic polynomials of degree not exceeding $n$. The constants $d_n$ and $c_n$ are identified as the smallest eigenvalues of certain Jacobi matrices and the two-sided estimates for $d_n$ and $c_n$ of the form $$ 4-\frac{c}{\ln n}< d_n, c_n<4-\frac{c}{\ln^2 n}\,,\qquad c>0\, $$ are established.
We present results of an experiment where, using a 200 GW CO2 laser seed, a 65 MeV electron beam was decelerated down to 35 MeV in a 54 cm long strongly tapered helical magnetic undulator, extracting over 30$\%$ of the initial electron beam energy to coherent radiation. These results demonstrate unparalleled electro-optical conversion efficiencies for a relativistic beam in an undulator field and represent an important step in the development of high peak and average power coherent radiation sources.