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ok, what about the question with which iv'e opened the thread?
A question on sequence of functions.
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SUMMARY
The discussion centers on the existence of a continuous function g: R -> R that is not periodic, yet allows for every sequence of real numbers {a_k} to produce a sequence of functions g_k(x) = g(x + a_k) with a uniformly converging subsequence on R. Participants propose various functions, including g(x) = sin(e^x) and g(x) = exp(-x^2), while examining the implications of the Arzelà-Ascoli theorem. Ultimately, the consensus suggests that functions like g(x) = sin(x) + cos(πx) exemplify the desired properties, as they are continuous, bounded, and exhibit equicontinuity.
PREREQUISITES- Understanding of continuous functions and their properties
- Familiarity with the Arzelà-Ascoli theorem
- Knowledge of equicontinuity and uniform convergence
- Basic concepts of periodic and almost periodic functions
- Study the Arzelà-Ascoli theorem in detail
- Explore examples of almost periodic functions
- Investigate the properties of equicontinuous functions
- Learn about the implications of uniform convergence in functional analysis
Mathematicians, students of real analysis, and anyone interested in functional analysis and the properties of continuous functions.