Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description of a trivial analytic multifunction.

A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f₁,...,f_k:I\to ℝ$, k ≥ 2, denoted by $A^{[f₁,...,f_k]}$, is considered. Some properties of $A^{[f₁,...,f_k]}$, including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For a sequence of...

For various $L^p$-spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in $L^p\left(\mu \right)$. For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...

The basic tool considered in this paper is the so-called graded set, defined on the analogy of the family of α-cuts of a fuzzy set. It is also considered the corresponding extensions of the concepts of a point and of a real number (again on the analogy of the fuzzy case). These new graded concepts avoid the disadvantages pointed out by Gerla (for the fuzzy points) and by Kaleva and Seikkala (for the convergence of sequences of fuzzy numbers).

The key result (Theorem 1) provides the existence of a holomorphic approximation map for some space of C∞-functions on an open subset of Rn. This leads to results about the existence of a continuous linear extension map from the space of the Whitney jets on a closed subset F of Rn into a space of holomorphic functions on an open subset D of Cn such that D ∩ Rn = RnF.

In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties: $1_{}$. if M is a linear manifold, then (M) contains convex functions, $2_{}$. (·) is invariant under diffeomorphisms, $3_{}$. each f ∈ (M) is differentiable on a dense $G_{\delta }$-set, is investigated.

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family $F^t:t\ge 0$ of continuous linear set-valued functions $F^t:K\to cc\left(K\right)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $\Phi (t,x)=F^t\left(x\right)$ is a solution of the problem $D_t\Phi (t,x)=\Phi (t,G\left(x\right)):=\bigcup \Phi (t,y):y\in G\left(x\right)$, Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_t\Phi (t,x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G\left(x\right):=li{m}_{s\to 0+}(F^s\left(x\right)-x)/s$ for x ∈ K.