The multifractal formalism for single measure is reviewed. Next, a mixed generalized multifractal formalism is introduced which extends the multifractal formalism of a single measure based on generalizations of the Hausdorff and packing measures to a vector of simultaneously many measures. Borel-Cantelli and Large deviations Theorems are extended to higher orders and thus applied for the validity of the new variant of the multifractal formalism for some special cases of multi-doubling type measures.

Let's take $H$ as an infinite-dimensional Hilbert space and $K(H)$ be the set of all compact operators on $H$. Using Spectral theorem for compact self-adjoint operators, we prove the Hyers-Ulam stability of Jensen $*$-derivations from $K(H)$ into $K(H)$.

This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract framework of Fourier analysis for the Hilbert function space $L^2(G/H,\mu)$.

In this survey the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the $L^p$ spaces and recent results in Orlicz spaces. The notion of balanced point, introduced by Chui et al. in 1984 is extensively used.

We establish several fundamental identities, including recurrence relations, degree elevation formulas, partition of unity and Marsden identity, for quantum Bernstein bases and quantum Bézier curves. We also develop two term recurrence relations for quantum Bernstein bases and recursive evaluation algorithms for quantum Bézier curves. Our proofs use standard mathematical induction and other elementary techniques.

The operator equation $\lambda M_{\overline z} X = X M_{z^k}$, for $k \geq 2, $ $\lambda \in \mathbb{C}$, is completely solved. Further, some algebraic and spectral properties of the solutions of the equation are discussed.

Here we consider the following strongly singular integral $$T_{\Omega,\gamma,\alpha,\beta}f(x,t)=\int_{R^n} e^{i|y|^{-\beta}}\frac {\Omega(\frac{y}{|y|})}{|y|^{n+\alpha}}f(x-y,t-\gamma(|y|))dy,$$ where $\Omega\in L^p(S^{n-1}),$ $ p>1,$ $n>1,$ $\alpha>0$ and $\gamma$ is convex on $(0,\infty)$.
We prove that there exists $A(p,n)>0$ such that if $\beta>A(p,n)(1+\alpha)$, then $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself and the constant is independent of $\gamma$. Furthermore, when $\Omega\in C^\infty(S^{n-1})$, we will show that $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself only if $\beta>2\alpha$ and the constant is independent of $\gamma$.