We will study the strong approximation by Fourier-Vilenkin series using matrices with some general monotone condition. The strong Vallee-Poussin, which means of Fourier-Vilenkin series is also investigated.

In this paper, by applying the technique of the sharp maximal function and the equivalent representation of the norm in the Lebesgue spaces with variable exponent, the boundedness of the parameterized Littlewood-Paley operators, including the parameterized Lusin area integrals and the parameterized Littlewood-Paley $g_{\lambda}^{\ast}$-functions, is established on the Lebesgue spaces with variable exponent. Furthermore, the boundedness of their commutators generated respectively by BMO functions and Lipschitz functions are also obtained.

We establish the construction theory of function based upon a local field $K_p$ as underlying space. By virtue of the concept of pseudo-differential operator, we introduce "fractal calculus" (or, $p$-type calculus, or, Gibbs-Butzer calculus). Then, show the Jackson direct approximation theorems, Bernstein inverse approximation theorems and the equivalent approximation theorems for compact group $D(\subset K_p)$ and locally compact group $K^+_p(=K_p)$, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the Hölder-type space $C^\sigma(K_p), $ $\sigma>0$, are proved; then the equivalent approximation theorem on Sobolev-type space $W^r_\sigma(K_p),$ $ \sigma\geq 0,$ $ 1\leq r<+\infty$, is shown.

In the present paper, we propose the $q$ analogue of Szász-Beta-Stancu operators. By estimate the moments, we establish direct results in terms of the modulus of smoothness. Investigate the rate of point-wise convergence and weighted approximation properties of the $q$ operators. Voronovskaja type theorem is also obtained. Our results generalize and supplement some convergence results of the $q$-Szász-Beta operators, thus they improve the existing results.

In this paper, we introduce a condition weaker than the $L^p$ differentiability, which we call $C^p$ condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be $L^p$ differentiable. In addition, we study the convexity of  the set of cluster points of the net of best approximations of $f$, $\{P_\epsilon(f)\}$ as $\epsilon \to 0$.

In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrödinger operator on Morrey spaces with variable exponent.

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|\leq k$. For $k=1$, it is known that for each $r> 0$ and $|\alpha|\geq 1$, $$n(|\alpha|-1)\Big\{\int_{0}^{2\pi}|P(e^{i\theta})|^{r}d\theta\Big\}^{\frac{1}{r}}\leq \Big\{\int_{0}^{2\pi}|1+e^{i\theta}|^{r}d\theta\Big\}^{\frac{1}{r}}\max_{|z|=1}\big|D_{\alpha}P(z)\big|.$$ In this paper, we shall first consider the case when $k\geq 1$ and present certain generalizations of this inequality. Also for $k\leq 1$, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.