Evaluate limit of this integral using positive summability kernels

In summary, the process of evaluating the limit of an integral using positive summability kernels involves applying specific mathematical techniques to approximate the integral through a sequence of simpler functions. This approach utilizes properties of positive kernels to ensure convergence and facilitate the analysis of the integral's behavior as parameters change. By leveraging these kernels, one can derive meaningful results and insights regarding the integral's limit, often leading to a clearer understanding of its underlying structure and characteristics.
  • #1
psie
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Homework Statement
Let ##\varphi## be defined by ##\varphi(x)=\frac{15}{16}(x^2-1)^2## for ##|x|<1## and ##\varphi(x)=0## otherwise. Let ##f## be a function with a continuous derivative. Find the limit $$\lim_{n\to\infty}\int_{-1}^1n^2\varphi '(nx)f(x)dx.$$
Relevant Equations
Positive summability kernels, see e.g. Wikipedia.
Integrating the integral by parts, using that the antiderivative of ##\varphi'(nx)## is ##\frac1{n}\varphi(nx)##, I get
$$\big[n\varphi(xn)f(x)\big]_{-1}^1-\int_{-1}^1 n\varphi(nx)f'(x)dx=0-\int_{-1}^1 n\varphi(nx)f'(x)dx.$$ I used the fact that ##\varphi(n)## and ##\varphi(-n)## both equal ##0##, since ##n\geq 1##.

However, I'm stuck here. This is a problem from a section on positive summability kernels, but I have been unable to verify what the kernel is in this exercise, if there is any. Appreciate any help.
 
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  • #2
I solved it I think. ## n\varphi(nx)## is a positive summability kernel and the integral therefor evaluates to ##-f'(0)##.
 
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  • #3
You could try checking by plugging in ##f(x)=x^r##.
 
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FAQ: Evaluate limit of this integral using positive summability kernels

What are positive summability kernels?

Positive summability kernels are functions used to approximate the identity operator in the context of integration. They are typically non-negative, integrable functions that converge to a Dirac delta function as a parameter tends to zero. These kernels are used to smooth or approximate other functions.

How do positive summability kernels help in evaluating the limit of an integral?

Positive summability kernels help in evaluating the limit of an integral by smoothing the integrand and making it easier to handle analytically. As the parameter of the kernel approaches zero, the integral of the product of the kernel and the function approximates the value of the function at a specific point, facilitating the evaluation of the limit.

What is an example of a positive summability kernel?

An example of a positive summability kernel is the Gaussian kernel, given by \( K_\epsilon(x) = \frac{1}{\sqrt{2\pi\epsilon}} e^{-\frac{x^2}{2\epsilon}} \). As \(\epsilon\) approaches zero, this kernel converges to the Dirac delta function, which is useful for approximating integrals.

How do you set up an integral with a positive summability kernel?

To set up an integral with a positive summability kernel, you typically multiply the function you are integrating by the kernel and then integrate over the entire domain. For example, if \( f(x) \) is the function and \( K_\epsilon(x) \) is the kernel, you would evaluate \( \int_{-\infty}^{\infty} f(x) K_\epsilon(x - a) \, dx \), where \( a \) is the point at which you want to evaluate the function.

What are the conditions for a kernel to be considered a positive summability kernel?

For a function \( K_\epsilon(x) \) to be considered a positive summability kernel, it must satisfy the following conditions: (1) Non-negativity: \( K_\epsilon(x) \geq 0 \) for all \( x \); (2) Normalization: \( \int_{-\infty}^{\infty} K_\epsilon(x) \, dx = 1 \); and (3) Convergence to the Dirac delta function: As \(\epsilon\) approaches zero, \( K_\epsilon(x) \) should converge to \( \delta(x) \), the Dirac delta function.

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