This paper is concerned with Korovkin type approximation theorems. We characterize $BKW$-operators on the Banach space of real valued continuous functions on the unit interval for the test functions $\{1,t,t^2,t^3\}$. It is also investigated when subtraction of composition operators are $BKW$-operators for $\{1,t,t^2,t^3,t^4\}$.

The small Kakeya maximal operator, $M_{a,N}$, in $\mathbf{R}^d$ is defined by averages on cylinders with the width $a$ and the height $Na$. We show that the inequality $\lVert M_{a,N}f\rVert_d \leq C\log N\lVert f\rVert_d$ holds for the functions of square radialy type, where $C$ is a constant depending only on $d$.

We consider the equation \[ \frac{\partial u}{\partial t}(t,x)=-\Delta^{2}u(t,x) \] for the biharmonic operator $-\Delta^2$. We define the pseudo process corresponding to this equation as Nishioka's sense. We obtain the Laplace-Fourier transform of the joint distribution of the first hitting time $\tau(\omega)=\inf\{t>0:\omega(t)<\alpha t-a\}$ $(a>0, \alpha\in\mathbf{R})$ and the first hitting place $\omega(\tau)$, where each path $\omega(t)$ starts from 0 at $t=0$.

We show that the algebraic unknotting number of a classical knot $K$, defined by Murakami [9], is equal to the minimum number of unknotting operations necessary to transform $K$ to a knot with trivial Alexander polynomial. Furthermore, we define a new operation, called an elementary twisting operation, for smooth $(2n-1)$-knots with $n\geq 1$ and odd, and show that this is an unknotting operation for simple $(2n-1)$-knots. Moreover, the unknotting number of a simple $(2n-1)$-knot defined by using the elementary twisting operation is equal to the algebraic unknotting number of the $S$-equivalence class of its Seifert matrix if $n\geq 3$.

There are two types of generalizations of selfdecomposability of probability measures on $\mathbf{R}^d, d\geq 1$ : the $c$-decomposability and the $C$-decomposability of Loève and Bunge on the one hand, and the semi-selfdecomposability of Maejima and Naito on the other. The latter implies infinite divisibility but the former does not in general. For $d\geq 2$ introduction of operator (matrix) normalizations yields four kinds of classes of distributions on $\mathbf{R}^d : L_{0}(b,Q),\tilde{L}_{0}(b,Q),L_{0}(C,Q)$, and $\tilde{L}_{0}(C,Q)$, where $0<b<1$, $Q$ is a $d\times d$ matrix with eigenvalues having positive real parts, and $C$ is a closed multiplicative subsemigroup of $[0,1]$ containing 0 and 1. Further, each of these classes generates the Urbanik-Sato type decreasing sequence of its subclasses. Characterizations and relations of these classes and subclasses are established. They complement and generalize results of Bunge, Jurek, Maejima and Naito, and Sato and Yamazato.