- #1
psie
- 108
- 10
- 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.
$$\big[n\varphi(xn)f(x)\big]_{-1}^1-\int_{-1}^1 n\varphi(nx)f'(x)dx=0-\int_{-1}^1 n\varphi(nx)f'(x)dx.$$ 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.