Related papers: A generalisation of the Babbage functional equatio…

Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem. {\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, then $g=0$. There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that…

The theory of linear Fredholm integral-functional equations of the second kind with linear functionals and with a parameter is considered. The necessary and sufficient conditions are obtained for the coefficients of the equation and those…

We characterise all pairs of finite order entire functions whose magnitudes agree on two arbitrary lines in the complex plane by means of the Hadamard factorisation theorem. Building on this, we also characterise all pairs of second order…

We consider the equation $$ ab + cd = \lambda, \qquad a\in A, b \in B, c\in C, d \in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \subseteq F_q$. The question of solvability of such and…

The functionals on an ordered semigroup S in the category Cu--a category to which the Cuntz semigroup of a C*-algebra naturally belongs--are investigated. After appending a new axiom to the category Cu, it is shown that the "realification"…

We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)= \sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in {GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued functions and in the…

We prove that a $C^{\infty}$ semialgebraic local diffeomorphism of $\mathbb{R}^n$ with non-properness set having codimension greater than or equal to $2$ is a global diffeomorphism if $n-1$ suitable linear partial differential operators are…

Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1) there is an algorithm that for every computable function f:N-->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any…

Carleson's Theorem asserts the pointwise convergence of Fourier series of square integrable functions. We give a complete proof, following joint work of the author and C. Thiele. Over 20 exercises are also detailed. We also discuss the…

We describe an explicit connection between solutions to equations $Df=0$ (the Generalized Cauchy-Riemann equation) and $(D+M)f=0$, where operators $D$ and $M$ commute. The described connection allows to construct a ``function theory'' (the…

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable…

A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the…

For a subfield K of C, we denote by C^K the category of algebras of functions defined on the globally subanalytic sets that are generated by all K-powers and logarithms of positively-valued globally subanalytic functions. For any function f…

It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we…

We prove a $C^{1,\alpha}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity…

Every real Bank-Laine function of finite order, whose zeros are all real but neither bounded above nor bounded below, either has an explicit representation in terms of trigonometric functions or has zeros with exponent of convergence at…

We present an algorithm producing all rational functions $f$ with prescribed $n+1$ Taylor coefficients at the origin and such that $\|f\|_\infty\le 1$ and $\deg f\le k$ for every fixed $k\ge n$. The case where $k<n$ is also discussed.

We prove, using a fixed point theorem in a Banach algebra, an existence result for a fractional functional differential equation in the Riemann-Liouville sense. Dependence of solutions with respect to initial data and an uniqueness result…