Related papers: Elementary analytic functions in $VTC^0$
A class of rational functions characterized by some wonderful properties is studied. The properties that identify this class include simple algebra (their inverses can be expressed in radicals), simple topology (the total space of the…
The elementary arithmetic operations $+,\cdot,\le$ on integers are well-known to be computable in the weak complexity class $\mathrm{TC}^0$, and it is a basic question what properties of these operations can be proved using only…
We comment on recent results in the field of information based complexity, which state (in a number of different settings), that approximation of infinitely differentiable functions is intractable and suffers from the curse of…
In this article we define restricted log-exp-analytic functions as compositions of log-analytic functions and exponentials whose logarithm are locally bounded. We prove that the derivative of a restricted log-exp-analytic function is again…
We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The…
A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$h: \mathbb{R} \to \mathbb{R}, x \mapsto \left\{\begin{array}{ll} x^r, & x > 0, \\ 0, & \textnormal{ else, }…
We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to…
We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application…
The class of uniformly computable real functions with respect to a small subrecursive class of operators computes the elementary functions of calculus, restricted to compact subsets of their domains. The class of conditionally computable…
This is a continuation of our earlier paper \cite{PT3}. We consider here operator-valued functions (or infinite matrix functions) on the unit circle $\T$ and study the problem of approximation by bounded analytic operator functions. We…
A fast approximation to the Boys functions (related to the lower incomplete gamma function of half-integer parameter) by a single closed-form analytical expression for all argument values have been developed and tested. Besides the…
This paper lays down a foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theory based on the representation theory of SL(2,R) group. We describe here geometries of…
One of the elegant achievements in the history of proof theory is the characterization of the provably total recursive functions of an arithmetical theory by its proof-theoretic ordinal as a way to measure the time complexity of the…
We show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\R^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex…
For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These…
We consider a Froeschl\'e map and we add a weak dissipation of the form $\lambda(\varepsilon) = 1- \varepsilon^3$, where $\varepsilon$ is the parameter of perturbation. We compute formal expansions of lower dimensional tori, both in the…
The function $P(T)=\sum_{i=0}^\infty c_i T^i$ is admissible if $c_i\geq 0$, $\sum_{i=0}^\infty c_i\leq 1$. For any given set of admissible functions $P_1,\dots, P_k$ there is a unitary operator $T$ of dynamic origin such that the weak…
A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are…
We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic…
We present a method for constructing global analytical expressions that approximate a function over its entire range. These approximations not only mirror the original function as accurately as desired, but are purposefully created to…