Narrow Results

In this paper, we mainly make research on whether fractal characteristics of an object with a fractal curve shape will keep invariable under being subjected to horizontal linear stretching. For two continuous functions $f(x)$ and $g(x)$, it has been proven that the graph of $f[g(x)]$ has the same fractal dimensions with the graph of $f(x)$ when $g(x)$ is a linear function. Some examples have also been provided to corroborate our theoretical results. This work may contribute to the dimensional theory of fractal curves and have practical application significance in physics.

Let X ⊂ ℝ be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e. the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the strength of the formula by proving various corollaries. We conclude by constructing a collection of sets X with infinitely many isolated points, having upper box dimension a taking values from zero to one while their graph box dimension takes any value in [max{2a, 1}, a + 1], answering this way, negatively to a conjecture posed.¹

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

In this paper, we make research on the fractal structure of the space of continuous functions and explore how the fractal dimension of continuous functions under certain operations changes. We prove that any nonzero real power and the logarithm of a positive continuous function can keep the fractal dimensions of bi-Lipschitz invariance closed. For continuous functions having finite zero points, the relationship between its global behavior and the local behavior of its square on zero points has been given. Further, we discuss the fractal dimension of the product of continuous functions and provide the product decomposition of a continuous function in terms of the lower and upper box dimensions. Some special properties of the space of one-dimensional continuous functions have also been shown.

This paper investigates the variation of fractal dimensions of continuous functions under operations. It is found that the Box dimension of the logarithm of positive continuous functions remains constant, and any nonzero real power of a positive continuous function can keep fractal dimensions variation closed. The study also discusses the fractal dimension of logarithmic function with bases of fractal functions, and the corresponding fractal dimension is determined by the function with the larger fractal dimension.

In this paper, we explore upper box dimension of continuous functions on $[0,1]$ and their Riemann–Liouville fractional integral. Firstly, by comparing function limits, we prove that the upper box dimension of the Riemann–Liouville fractional order integral image of a continuous function will not exceed $2-\upsilon$, the result similar to [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Appl. Sin. E 32 (2016) 1494–1508]. Secondly, we prove that upper box dimension of multiple algebraic sums of continuous functions does not exceed the largest box dimension among them, backing up our conclusion with an appropriate example. Finally, we draw the same conclusions for the product of multiple continuous functions.

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

In [Integrable functions for Bernoulli measures of rank 1, Ann. Math. Blaise Pascal17 (2010) 349–363; Mesures p-adiques et suites classiques de nombres. Thèse de Doctorat, Université de Bamako, (2011)] we have begun the study of non-Archimedean integrable functions with respect to the normalized Bernoulli measures of rank 1, denoted by μ_1,α. The integrability is within the theory of non-Archimedean integration due to Monna and Springer [Intégration non-archimédienne. II, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math.25 (1963) 643–653; Intégration non-archimédienne. I, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math.25 (1963) 634–642], which we will call the Monna–Springer theory. In this paper which contains strictly the results of [Integrable functions for Bernoulli measures of rank 1, Ann. Math. Blaise Pascal17 (2010) 349–363; Mesures p-adiques et suites classiques de nombres. Thèse de Doctorat, Université de Bamako (2011)], we cover the integrability with respect to all normalized Bernoulli measures of rank 1. We show that, the integrable functions are reduced to the continuous functions when α is a p-adic unit different from 1 and -1, and all p-adic functions are integrable for α ∈ {-1, 1}. Again following the Monna–Springer theory, we also show that for the measure μ(5) = ∑_α⁴=1 μ_1,α, the space of integrable functions is equal to the space of continuous functions.