论文标题
具有加权的Takagi-Landsberg功能的分数积分，衍生物和积分方程
Fractional integrals, derivatives and integral equations with weighted Takagi-Landsberg functions
论文作者
论文摘要
在本文中，我们发现了Takagi-Landsberg功能的分数Riemann-Liouville衍生物。此外，我们介绍了他们的概括，称为加权的高田堡功能，这些功能在Schauder的扩展中具有任意界限的系数。 $ [0,1] $上$ h> 0 $的加权landsberg的类别与$ h $-hölder连续功能相吻合$ [0,1] $。基于HAAR和SCHAUDER函数的计算分数积分和衍生物，我们获得了Hölder连续函数的分数衍生物的新系列表示。该结果允许获得Riemann-Stieltjes积分的新公式。这种系列表示的应用是Volterra的数值解和由Hölder连续函数驱动的线性积分方程的新方法。
In this paper we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg functions which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of the weighted Takagi- Landsberg functions of order $H>0$ on $[0,1]$ coincides with the $H$-Hölder continuous functions on $[0,1]$. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a Hölder continuous function. This result allows to get the new formula of a Riemann-Stieltjes integral. The application of such series representation is the new method of numerical solution of the Volterra and linear integral equations driven by a Hölder continuous function.
