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Sequences and Series of Functions
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
As we have seen, there are two natural definitions of convergence for sequences of functions. Uniform convergence is a stronger condition than pointwise convergence, in the sense that every uniformly convergent sequence of functions is also pointwise convergent, but the converse is not true. A question of interest is which properties of the functions in our sequence carry through the limit under each kind of convergence. A major example is continuity: we have already seen that a sequence of continuous functions (fn)n=1∞ can converge to a discontinuous function f, provided that convergence is pointwise convergence. Surprisingly, this cannot happen in the case of uniform convergence. The following theorem guarantees that the uniform limit of continuous functions is a continuous function.
Preliminaries
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
A sequence of real or complex-valued functions {fn(x)}converges uniformlyConvergenceuniform to a function f(x) on a set S in Euclidean space if given any ε>0 there is a positive integer N, dependent only on ɛ and the set S, such that |fn(x)−f(x)|<ε for all x in S when n > N. The distinction between pointwise convergence and uniform convergence is that when the convergence is uniform once ɛ is given a single N can be found that works simultaneously for all x in S.
Orthogonal Expansions
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
The estimate (10.4.8 can be used in a comparison test to show that the Fourier series is absolutely dominated by a p‐series of the form ∑n-p $ \mathop \sum \limits_{{}}^{{}} n^{ - p} $ , which converges for p > 1. A condition that f’ is a piecewise continuous function is sufficient: it is known that for any discrete number of points, there exists a continuous function for which its Fourier series diverges at this set of points. In 1903, Lebesgue $ {\text{Lebesgue}} $ 10 proved that the Fourier coefficients an and bn of a Lebesgue‐or Riemann‐integrable function approach 0 as n → ∞. Andrew Kolmogorov gave an example in 1929 of an integrable function whose Fourier series diverges at every point. For uniform convergence of the Fourier series, we need to impose an additional condition on the function.
Descent methods with computational errors in Banach spaces
Published in Optimization, 2020
Simeon Reich, Alexander J. Zaslavski
Given a Lipschitz convex and coercive objective function on a Banach space, we consider a complete metric space of vector fields, which are self-mappings of the Banach space, with the topology of uniform convergence on bounded subsets. With each such vector field, we associate a certain iterative process. In our previous work [1,2] we introduced the class of regular vector fields and showed, using the generic approach and the porosity notion, that a typical vector field is regular and that for a regular vector field, the values of the objective function at the points generated by our process tend to its infimum. Taking into account computational errors, we study in the present paper the behaviour of the values of the objective function for the process generated by a regular vector field and show that if the computational errors are small enough, then the values of the objective functions become close to its infimum.
REGINN-IT method with general convex penalty terms for nonlinear inverse problems
Published in Applicable Analysis, 2022
In this subsection, the regularization property of Algorithm 3.2 will be analysed. To this end, we first show the uniform convergence result of the sequence . This is crucial for the proof of regularization property. Then, the stability result of Algorithm 3.2 is established. Finally, the regularization of Algorithm 3.2 follows from its stability and Theorem 3.9.
Simplified Levenberg–Marquardt method in Banach spaces for nonlinear ill-posed operator equations
Published in Applicable Analysis, 2023
Pallavi Mahale, Farheen M. Shaikh
In the Theorem 4.1, we have proved convergence of every sequence in . Next, we show uniform convergence result for all sequence in . For proving the uniform convergence result, we need to prove two preliminary results. First one in this row is as follows: