One reminds for all function f : n → the so-called (min, +)-wavelets which are lower and upper hulls build from (min, +) analysis [12, 13]. One shows that this analysis can be applied numerically to the Weierstrass and Weierstrass-Mandelbrot functions, and that one recovers their theoretical Hölder exponents and fractal dimensions.

An elegant expression is obtained for the product of the inverse Weierstrass-Laguerre transforms of two functions in terms of their convolution. It is also shown how the main result can be extended to hold for the product of the inverse Weierstrass-Laguerre transforms of several functions.

O objetivo desta dissertação é demonstrar e aplicar o Teorema da Aproximação de Weierstrass, sobre aproximação de funções contínuas em intervalos fechados e limitados da reta por polinômios, e o Teorema de Stone-Weierstrass, sobre aproximação de funções contínuas definidas em espaços topológicos compactos. Como aplicações do Teorema da Aproximação de Weierstrass tratamos o problema dos momentos de uma função contínua e a aproximação de funções contínuas definidas na reta por funções infinitamente diferenciáveis. Como aplicações do Teorema de Stone-Weierstrass provamos que o espaço C(K) das funções contínuas no compacto K é separável se e somente se K é metrizável e também a existência de um compacto K tal que C(K) é isometricamente isomorfo ao espaço `1 das sequências limitadas. __________________________________________________________________________________________ ABSTRACT; The aim of this dissertation is to prove and apply the Weierstrass Approximation Theo- rem, on the approximation of continuous functions on bounded closed intervals by polyno- mials, and the Stone-Weierstrass Theorem, on the approximation of continuous functions on compact topological spaces. As applications of the Weierstrass Approximation Theo- rem we deal with the momentum problem for continuous functions and the approximation of continuous functions on the line by in¯nitely di®erentiable functions. As applications of the Stone-Weierstrass Theorem we prove that the space C(K) of continuous functions on the compact K is separable if and only if K is metrizable and the existence of a com- pact space K such that C(K) is isometrically isomorphic to the space `1 of bounded sequences.; Dissertação (mestrado)-Universidade Federal de Uberlândia...

We classify the computational content of the Bolzano-Weierstrass Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem of that space. By specialization to sequences with a relatively compact range we obtain a characterization of the Bolzano-Weierstrass Theorem as the derivative of compact choice. In particular, this shows that the Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the jump of the lesser limited principle of omniscience LLPO and the Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump of the idempotent closure of LLPO. We also introduce the compositional product of two Weihrauch degrees f and g as the supremum of the composition of any two functions below f and g, respectively. We can express the main result such that the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's Lemma and the Monotone Convergence Theorem. We also study the class of weakly limit computable functions...

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.; Comment: 20 pages

In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein constructed a simple (non-holomorphic) completion of this form which is doubly periodic. This theorem has begun to play an important role in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as well as Alfes, Griffin, Ono, and the author. In particular, this simple completion of $\zeta$ provides a powerful method to construct harmonic Maass forms of weight zero which serve as canonical lifts under the differential operator $\xi_{0}$ of weight 2 cusp forms, and this has been shown in to have deep applications to determining vanishing criteria for central values and derivatives of twisted Hasse-Weil $L$-functions for elliptic curves. Here we offer a new and motivated proof of Eisenstein's theorem, relying on the basic theory of differential operators for Jacobi forms together with a classical identity for the first quasi-period of a lattice. A quick inspection of the proof shows that it also allows one to easily construct more general non-holomorphic elliptic functions.; Comment: 3 pages...

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by K.Weierstrass and F.Klein was aimed to extend to higher genera Weierstrass theory of elliptic functions based on the Weierstrass $\sigma$-functions. Our development was motivated by the recent achievements of mathematical physics and theory of integrable systems that were based of the results of classical theory of multi-dimensional theta functions. Both theta and sigma-functions are integer and quasi-periodic functions, but worth to remark the fundamental difference between them. While theta-function are defined in the terms of the Riemann period matrix, the sigma-function can be constructed by coefficients of polynomial defining the curve. Note that the relation between periods and coefficients of polynomials defining the curve is transcendental. Since the publication of our 1997-review a lot of new results in this area appeared (see below the list of Recent References), that promoted us to submit this draft to ArXiv without waiting publication a well-prepared book. We complemented the review by the list of articles that were published after 1997 year to develop the theory of $\sigma$-functions presented here. Although the main body of this review is devoted to hyperelliptic functions the method can be extended to an arbitrary algebraic curve and new material that we added in the cases when the opposite is not stated does not suppose hyperellipticity of the curve considered.; Comment: 267 pages...

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we define the fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The Holder exponent and Box dimension of this function are calculated here. It is established that the Holder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function, independent of incorporating the fractional trigonometric function. This is new development in generalizing the classical Weierstrass function by usage of fractional trigonometric function and obtain its character and also of fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, and establishing that roughness index are invariant to this generalization.; Comment: 17 pages, 2 figures, submitted to Physics Letters A

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.; Comment: 19 pages. We have extended the Introduction, corrected some typos and tidied up some proofs, and inserted extra material on genus 3 curves

In this paper we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both $M/r_0$ and $M/b$ where $M$ is the mass of the black hole, $r_0$ the distance of closest approach of the light ray and $b$ the impact parameter. We also use the Weierstrass function formalism to analyze other more exotic cases such as Reissner-Nordstr\om null geodesics and Schwarzschild null geodesics in 4 and 6 spatial dimensions. Finally we apply Weierstrass functions to describe the null geodesics in the Ellis wormhole spacetime and give an analytic expansion of the deflection angle in $M/b$.; Comment: Latex file, 19 pages 4 figures references and two comments added

We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the incremental quotients are bounded has maximal Hausdorff dimension.

We obtain the explicit value of the Hausdorff dimension of the graphs of the classical Weierstrass functions, by proving absolute continuity of the SRB measures of the associated solenoidal attractors.; Comment: 42 pages

Let $ \tau : [0,1] \rightarrow [0,1] $ be a piecewise expanding map with full branches. Given $ \lambda : [0,1] \rightarrow (0,1) $ and $ g : [0,1] \rightarrow \mathbb{R} $ satisfying $ \tau ' \lambda > 1 $, we study the Weierstrass-type function \[ \sum _{n=0} ^\infty \lambda ^n (x) \, g (\tau ^n (x)), \] where $ \lambda ^n (x) := \lambda(x) \lambda (\tau (x)) \cdots \lambda (\tau ^{n-1} (x)) $. Under certain conditions, Bedford proved that the box counting dimension of its graph is given as the unique zero of the topological pressure function \[ s \mapsto P ((1-s) \log \tau ' + \log \lambda) . \] We give a sufficient condition under which the Hausdorff dimension also coincides with this value. We adopt a dynamical system theoretic approach which was originally used to investigate special cases including the classical Weierstrass functions. For this purpose we prove a new Ledrappier-Young entropy formula, which is a conditional version of Pesin's formula, for non-invertible dynamical systems. Our formula holds for all lifted Gibbs measures on the graph of the above function, which are generally not self-affine.

We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Gauss plane). We also find a canonical representation of maximal strongly regular space-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Lorentz plane). This allows us to describe locally the solutions of the corresponding natural partial differential equations.; Comment: 15 pages

A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity theory, brane, string and Rundall-Sundrum theories. The obtained algebraic equation was brought by means of a linear-fractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with $g_{2}$ and $g_{3}$ functions of a complex variable instead of the definite complex numbers (known from the usual arithmetic theory of elliptic functions and curves). The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case, and the case of the infinite point of the linear-fractional transformation was investigated. Some relations were found, which ensure the parametrization of the cubic equation in its general form with the Weierstrass function.; Comment: v.2; submitted to Journ.Math.Phys.(October 2001); Latex (Sci.Word,amsmath style), 77 pages, no figures, 4 appendixes; Sect.III rewritten for more clear derivation of the cubic algebraic equation; clarifying comments in Sect.VI and in the Introduction; new Sect.VII added;2 references corrected; acknowledgments added

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show thair linear independence for the numbers of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.; Comment: 13 pages

The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a spherical pendulum (Weierstrass), and the motion of a heavy symmetric top with one fixed point (Weierstrass). The problem of the planar pendulum, in fact, can be used to construct the general connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful tools in the undergraduate curriculum.; Comment: 17 pages, 20 figures

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass $\zeta$-functions encode the central $L$-values and $L$-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by H\"ovel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of $E$. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain $p$-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.; Comment: To appear in Research in Number Theory