Convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##

In summary, the conversation discusses a problem involving the convergence of a sequence of functions. The sequence, made up of powers of the sinc function, is shown to converge pointwise to the indicator function. However, it is uncertain if the convergence is uniform. It is mentioned that the uniform convergence theorem states that a continuous function must be continuous for the sequence to converge uniformly, but since the indicator function is not continuous, it can be concluded that the sequence does not converge uniformly.
  • #1
Wuberdall
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Hi Physics Forums,

I have a problem that I am unable to resolve.

The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to decide if the convergence is uniform or not.

I hope that someone in here can help me, either by providing a reference or a sketch proof.

Thanks in regards.
 
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  • #2
Wuberdall said:
Hi Physics Forums,

I have a problem that I am unable to resolve.

The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to decide if the convergence is uniform or not.

I hope that someone in here can help me, either by providing a reference or a sketch proof.

Thanks in regards.
What does the uniform convergence theorem say about continuous functions?
 
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  • #3
haruspex said:
What does the uniform convergence theorem say about continuous functions?
Thanks,

Each of the ##\mathrm{sinc}^n(x)## functions in the sequence are continuous. Thus IF the sequence where to converge uniformly to ##\mathbf{1}_{\{0\}}##, then ##\mathbf{1}_{\{0\}}## has to be continuous (which it is not). Consequently, the sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## is not uniformly converging.
 
  • #4
Wuberdall said:
Thanks,

Each of the ##\mathrm{sinc}^n(x)## functions in the sequence are continuous. Thus IF the sequence where to converge uniformly to ##\mathbf{1}_{\{0\}}##, then ##\mathbf{1}_{\{0\}}## has to be continuous (which it is not). Consequently, the sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## is not uniformly converging.
Quite so.
 
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1. What is the definition of convergence in mathematics?

The concept of convergence in mathematics refers to the idea that a sequence or series of numbers approaches a specific limit as the number of terms increases. In other words, as more terms are added to the sequence or series, the numbers get closer and closer to a specific value.

2. What is the significance of the ##\mathrm{sinc}^n(x)## function in convergence?

The ##\mathrm{sinc}^n(x)## function is often used as an example in discussing convergence because it demonstrates a unique behavior. As ##n## increases, the function approaches a specific limit, but does not actually reach it. This behavior is known as "convergence without reaching."

3. How is the convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## determined?

The convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## can be determined using various methods, such as the ratio test, the root test, or the comparison test. These tests involve analyzing the behavior of the function as ##n## increases and determining if it approaches a specific limit or not.

4. Does the value of ##x## affect the convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##?

Yes, the value of ##x## can affect the convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##. For some values of ##x##, the function may converge to a specific limit, while for others it may not converge at all. This is due to the behavior of the function as ##n## increases, which can be affected by the value of ##x##.

5. What are some real-world applications of studying the convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##?

The study of convergence in mathematics has many practical applications, such as in physics, engineering, and computer science. For example, understanding the convergence of series can help in analyzing the behavior of electric circuits, designing efficient algorithms, and predicting the behavior of physical phenomena. Additionally, the concept of convergence is also important in many areas of pure mathematics, such as analysis and number theory.

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