Related papers: Extension of functions with bounded finite differe…
Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables.
This paper constructs a class of non-integer dimensional continuous functions with one unbounded variation point, discusses their H\"older condition and variation on their domains. Specifically, the fractal dimension of a continuous…
A discrete Fourier analysis associated with translation lattices is developed recently by the authors. It permits two lattices, one determining the integral domain and the other determining the family of exponential functions. Possible…
We investigate properties of holomorphic extensions in the one-variable case of Whitney's Approximation Theorem on intervals. Improving a result of Gauthier-Kienzle, we construct tangentially approximating functions which extend…
We provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced. The boundary condition is allowed to be discontinuous and to depend…
A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…
We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire,…
A new scheme is proposed to construct an n-times differentiable function extension of an n-times differentiable function defined on a smooth domain D in d-dimensions. The extension scheme relies on an explicit formula consisting of a linear…
In this paper, new relations between the derivatives of the Legendre polynomials are obtained, and by these relations, new upper bounds for the Legendre coefficients of differentiable functions are presented. These upper bounds are sharp…
Index transforms with the product of the associated Legendre functions are introduced. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. The results are applied to solve a boundary value problem in a…
We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and…
The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions,…
It is shown how to define difference equations on particular lattices $\{x_n\}$, $n\in\mathbb{Z}$, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special…
This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the…
Given a convex set $C$ in a real vector space $E$ and two points $x,y\in C$, we investivate which are the possible values for the variation $f(y)-f(x)$, where $f:C\longrightarrow [m,M]$ is a bounded convex function. We then rewrite the…
We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions…
Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of analysis to investigate whether the function can be extended to all of $\mathbb{R}^m$ in…
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…
This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very…
We discuss two kinds of functorial prolongations of the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. We study the prolongation of vector fields in both cases…