Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

On the approximation by de la Vallée Poussin sums and interpolatory polynomials in Lipschitz norms

ПустьM ^m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x _k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$      

Lipschitz-Nikolski\imath constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM _n -operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$ and if for a functionf, the derivativeD ^m+2 f exist at a pointx∈(0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ where Ω is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$      

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.     

On the divisibility properties concerning sums of binomial coefficients

For any integer \(n> 1,\) we prove

$$\begin{aligned} 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(3k+1){2k\atopwithdelims ()k}^3(-8)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(6k+1){2k\atopwithdelims ()k}^3(-512)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(42k+5){2k\atopwithdelims ()k}^3 4096^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(20k^2+8k+1){2k\atopwithdelims ()k}^5(-4096)^{n-1-k}. \end{aligned}$$

The first three results confirm three divisibility properties on sums of binomial coefficients conjectured by Z.-W. Sun.

     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

Inequalities for sine sums with more variables

A result of Vietoris states that if the real numbers \(a_1,\ldots ,a_n\) satisfy

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

then, for \(x_1,\ldots ,x_m>0\) with \(x_1+\cdots +x_m <\pi \),

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$

We prove that \((**)\) (with “\(\ge \)” instead of “>”) holds under weaker conditions. It suffices to assume, instead of \((*)\), that

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$

and, moreover, \((**)\) is valid for a larger region, namely, \(x_1,\ldots ,x_m\in (0,\pi )\).

     

Inequalities for combinatorial sums

For \(k,l\in \mathbf {N}\), let

$$\begin{aligned}&P_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k-1} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }\\&\quad \text{ and }\quad Q_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }. \end{aligned}$$

We prove that the inequality

$$\begin{aligned} \frac{1}{4}\le P_{k,l} \end{aligned}$$

is valid for all natural numbers k and l. The sign of equality holds if and only if \(k=l=1\). This complements a result of Vietoris, who showed that

$$\begin{aligned} P_{k,l}<\frac{1}{2} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$

An immediate corollary is that

$$\begin{aligned} \frac{1}{4}\le P_{k,l}<\frac{1}{2} <Q_{k,l}\le \frac{3}{4} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$

The constant bounds are sharp.

     

Low Mach number limit of multidimensional steady flows on the airfoil problem

Given $$\alpha >0$$, we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$where $$W^{1,n}_{0,\mathrm{rad}}(B)$$ is the usual Sobolev spaces of radially symmetric functions on B in $${\mathbb {R}}^n$$ with $$n\ge 2$$. Without restricting to the class of functions $$W^{1,n}_{0,\mathrm{rad}}(B)$$, we should emphasize that the above inequalities fail in $$W^{1,n}_{0}(B)$$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.     

Inequalities for Alternating Trigonometric Sums

In 1970, J.B. Kelly proved that $$\begin{array}{ll}0 \leq \sum\limits_{k=1}^n (-1)^{k+1} (n-k+1)|\sin(kx)| \quad{(n \in \mathbf{N}; \, x \in \mathbf{R})}.\end{array}$$ We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums $$\begin{array}{ll} & \sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1) | \cos(kx) | \quad {\rm and}\\ & \quad{\sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1)\bigl( | \sin(kx) | + | \cos(kx)| \bigr)}.\end{array}$$      

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

Die Ungleichungen von Vietoris

The inequalities $$P_{k,l} = \frac{1}{{B(l + 1,k)}}\int\limits_0^{l/(k + l)} {x^l (1 - x)^{k - 1} dx = I_{l/(k + l)} (l + 1,k)< \frac{1}{2}}$$ and $$\Phi _{k,l,\mu } = \frac{1}{{B(k,k\mu + 1)}}\int\limits_0^1 {x^{k - 1} (1 - x)^{k\mu } I_x (l + 1,l\mu )dx< \frac{1}{2}}$$ are shown to be valid for any positive real numbersk, l, μ.     

On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $$p^2$$

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.

     

ON THE FIRST KIND OF RELATIVE k-JET COHOMOLOGY OF SINGULARITIES OF MAPGERMS

In this paper the author generalizes the computations about the first kind of k-jetcohomology in[5]to mapgerms.The main results are as follows:H~p(Ω_(,k-.,x))=0,0相似文献   

On Ivády’s bounds for the gamma function and related results

Periodica Mathematica Hungarica - We prove that the inequality $$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$ holds for all $$x\in [0,1]$$ , $$\beta \ge {\beta...     

Approximation by the Complex form of a Link Operator Between the Phillips and the Szász–Mirakjan Operators

Results in Mathematics - The link operator $${P_{\alpha}^\rho(f,x)=\sum_{k=1}^\infty s_{\alpha,k}(x)\int_0^\infty \theta_{\alpha,k}^\rho(t)f(t)dt+e^{-\alpha x}f(0)}$$ , $${\alpha, \rho &gt;...     

A remark on exceptional sets in Erdös-Rényi limit theorem

Let \((x_i)_{i=1}^{+\infty }\) be the digits sequence in the unique terminating dyadic expansion of \(x\in [0,1)\). The run-length function \(l_n(x)\) is defined by

$$\begin{aligned} l_n(x):=\max \left\{ j:x_{i+1}=x_{i+2}=\cdots =x_{i+j}=1\ \text {for some}\ 0\le i\le n-j\right\} . \end{aligned}$$

Erdös and Rényi proved that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=1, \text {a.e.}\ x\in [0,1). \end{aligned}$$

In this note, we show that for each pair of numbers \(\alpha ,\beta \in [0,+\infty ]\) with \(\alpha \le \beta \), the following exceptional set

$$\begin{aligned} E_{\alpha ,\beta }=\left\{ x\in [0,1):\liminf _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\alpha ,\ \limsup _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\beta \right\} \end{aligned}$$

has Hausdorff dimension one.

     

Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.