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Fourier Series
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Theorem 9.3.3. requires that we know the function f. Often, we only have a trigonometric series, and a test for uniform convergence is needed. While the Weierstrass M-test is quite powerful, it can be used only when the series is absolutely convergent as well. Exercise 9.3.7. shows that there are uniformly convergent series that do not converge absolutely. (However, see Exercise 9.3.23., for an unexpected converse.) The tests of Abel and Dirichlet (Section 8.3) were designed specifically for such situations.
Sequences and Series of Functions
Published in Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho, Concise Introduction to Basic Real Analysis, 2019
Hemen Dutta, Pinnangudi N. Natarajan, Yeol Je Cho
In an attempt to seek some simple ways of testing a series for uniform convergence without resorting to the definition in each case, we had the Weierstrass M-test introduced already. There are other tests that may be useful when the M-test is not applicable. One of these tests is as follows:
Analysis for two-dimensional inverse quasilinear parabolic problem by Fourier method
Published in Inverse Problems in Science and Engineering, 2021
by using Lipschitzs condition, we have By using the same operations we obtain: applying Gronwall's inequality to last inequality, we have The series which is consisting of the right hand side of (18) are convergent by ratio test. So, the series which is consisting of the left hand side of (18) are convergent by comparison test. Moreover, by the Weierstrass M test, the series is uniformly convergent.
Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition
Published in Applicable Analysis, 2019
The series which is consisting of the right-hand side of (12) are convergent by ratio test. So, the series which is consisting of the left-hand side of (12) are convergent by comparison test. Moreover, by the Weierstrass M test , the series is uniformly convergent. However, the general term of the sequence may be written as