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Integrals Depending on a Parameter
Published in John Srdjan Petrovic, Advanced Calculus, 2020
13.1.17. Suppose that, in addition to the hypotheses of Theorem 13.1.7, g is an absolutely integrable function on [a, b]. Prove that limt→t0∫abF(x,t)g(x)dx=∫abf(x)g(x)dx.
Fourier Series Expansions of Functions
Published in Wen L. Li, Weiming Sun, Fourier Methods in Science and Engineering, 2023
Since the Fourier coefficients for an absolutely integrable function approach to zero as m→∞, it is clear from (2.49): limm→∞mam=0.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
Definition 10.8: An absolutely integrable function f(x) is said to be piecewise monotone or satisfy Dirichlet's conditions on the interval [a,b] if:
Dimension reduction for k-power bilinear systems using orthogonal polynomials and Arnoldi algorithm
Published in International Journal of Systems Science, 2021
Zhen-Zhong Qi, Yao-Lin Jiang, Zhi-Hua Xiao
According to the expression of , it can be rewritten as the power series form: where are coefficients of . Conversely, polynomial can be expanded in terms of the orthogonal polynomial series as well Let matrices and , then it holds from (4) and (5). Furthermore, based on (3), the coefficients can be calculated by the equations as follows Xiao and Jiang (2016) with and . For an absolutely integrable function , it is expanded as the following form: and substitute (4) into (7). Comparing the coefficients of each power of t on the both sides, it leads to It is obvious that the coefficients in (7) are different from the coefficients of Taylor series for .
Fractional-order two-input two-output process identification based on Haar operational matrix
Published in International Journal of Systems Science, 2021
The M-square Haar matrix can be derived as (Li et al., 2015), The generalised operational matrix of fractional order integration (FOI) can be derived using basic orthogonal piecewise constant block pulse functions. The analytical expression of operational matrix of FOI for any absolutely integrable function can be derived using generalised operational matrix of FOI. The FOI of block pulse functions can be written as, where M shows number of elementary functions, denotes block pulse functions over the time interval and is called the generalised operational matrix of fractional integral given by where and .