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Suppose that $a<b$ are two fixed real numbers, and $g_n$ is a sequence of real functions on $[a,b]$.

What about good conditions approximately equivalent to the following proposition:

$$\lim_{n\to\infty}\int_a^bf(x)g_n(x)dx=0$$ for each $f\in\mathfrak R^2[a,b]$, where $\mathfrak R^2[a,b]$ means the existence of Riemann integrals $\int_a^b\lvert f(x)\rvert dx$ and $\int_a^b(f(x))^2dx$.

It's well-known that, if $g_n(x)=e^{2inx}$, then the preceding statement is true, for Riemann-Lebesgue lemma. Moreover, if $g(x)$ is a periodic function with period $T$, and $\int_0^T g(x)dx=0$, then the preceding statement is also true.

However, I'm thinking about a more general situation. I doubt it's related to linear transformations, and some theory about kernel functions, and about Silverman-Toeplitz theorem and the last post of mine. I need some detailed information and references.


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What is your space $R^2[a,b]$? Do you have further assumptions on $g_n$? Your convergence is something like "weak convergence" in function spaces. – gerw Jun 6 '13 at 20:07
@gerw That's about Riemann-integrability. See my edit. My goal is to merge the schema of Riemann-Lebesgue lemma and Toeplitz theorem. So far, I have no clear further assumptions. – Frank Science Jun 8 '13 at 9:19

This may be relevant. Suppose $\{K_\delta\}_{\delta>0}$ is a family of kernels that satisfies: for all $\delta > 0$

  1. $|K_\delta (x)|\le A\delta^{-d}$;
  2. $|K_\delta (x)|\le A\delta / |x|^{d+1}$;
  3. $\int K_\delta = 0$;

then $(f \ast K_\delta)(x) \rightarrow 0$ for a.e. $x$ as $\delta \rightarrow 0$ for every integrable function $f: \mathbb{R^d}\rightarrow \mathbb{C} $.

Here kernels $\{K_\delta\}$ approximate the delta function, but have average value 0 instead of 1.

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