Can All Smooth Functions Near Zero Be Expressed by This Double Series?

In summary, the conversation discusses the concept of a function being smooth but not analytic, with an example of ##\exp (-1/x^2)## near ##x = 0##. The question is raised about whether a double series, ##f(x) = \sum_{m = 0}^\infty \sum_{n = 0}^\infty a_m x^m \exp \bigg( - \frac{b_n}{x^n} \bigg)##, can express all smooth functions in an open neighborhood of 0. It is mentioned that this relates to instantons in QFT. However, it is concluded that ##f(x) = \exp ( - \exp (1/x
  • #1
Ben Niehoff
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As you should know, a function can be smooth in some neighborhood and yet fail to be analytic. A canonical example is ##\exp (-1/x^2)## near ##x = 0##. My question is this: suppose I want to express a given function as a double series,

[tex]f(x) = \sum_{m = 0}^\infty \sum_{n = 0}^\infty a_m x^m \exp \bigg( - \frac{b_n}{x^n} \bigg)[/tex]
Is this series general enough to express all smooth functions in some open neighborhood of 0? And if not, what do the counterexamples look like?

This has some relationship to instantons in QFT.
 
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  • #2
Never mind, I see that

[tex]f(x) = \exp ( - \exp (1/x^2) )[/tex]
is a counterexample.

Carry on, then.
 

FAQ: Can All Smooth Functions Near Zero Be Expressed by This Double Series?

What is a smooth function?

A smooth function is a mathematical function that has continuous derivatives of all orders. This means that the function is infinitely differentiable and has no sharp corners or breaks in its graph.

What is the difference between a smooth function and a non-smooth function?

The main difference between a smooth function and a non-smooth function is that a smooth function has continuous derivatives of all orders, while a non-smooth function does not. This results in a non-smooth function having sharp corners or breaks in its graph, while a smooth function has a smooth and continuous graph.

Why are smooth functions important?

Smooth functions are important in many areas of mathematics, physics, and engineering. They are used to model and describe real-world phenomena such as motion, growth, and changes in physical quantities. Additionally, smooth functions have many useful properties and allow for mathematical analysis and predictions.

How do you determine if a function is smooth?

To determine if a function is smooth, you need to check if it has continuous derivatives of all orders. This can be done by taking the derivative of the function and checking if it is continuous. If the derivative is continuous, you can then take the derivative of the derivative and repeat the process until you have checked for continuity in all orders.

Can all functions be smooth?

No, not all functions can be smooth. Some functions, such as step functions or absolute value functions, have discontinuities and therefore cannot have continuous derivatives of all orders. However, many functions that are commonly used in mathematics and science, such as polynomials and trigonometric functions, are smooth.

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