What are the conditions sufficient and necessary on $g(t)$ for the Dirichlet integral to be equal to $\frac{\pi}{2} g(0+)$?

Dirichlet Integral of a function $g\colon \mathbb{R} \to \mathbb{R}$ is defined as $$DI(\alpha) = \int_0^{\delta} g(t) \frac{\sin(\alpha t)}{t} dt$$ assume $\alpha \in \mathbb{N}$

For the equality $$\lim_{\alpha \to \infty} DI(\alpha) = \frac{\pi}{2} g(0+)$$ to hold, what are the necessary and sufficient condition required on $g(t)$ ?

Jordan's test tells us that the above mentioned equality holds when $g(t)$ is of bounded variation in $(0,\delta)$. Dini's test tells us that the equality holds when the below mentioned integral exists. $$\int_0^{\delta}\frac{g(t)-g(0+)}{t}dt$$.

I have read from books that there are functions which satisfy Jordan's test but not Dini's and functions which satisfy Dini's test but not Jordan's test. Both these tests are giving sufficient conditions.

Hence my question is, are there any conditions which are necessary and sufficient, for the above equality to hold ? Please give some references to any.

• What does $g(0+)$ mean? – Ryan Budney Mar 25 '12 at 2:59
• @Ryan : $g(0+) = \lim_{t\to 0+} g(t)$ – Rajesh Dachiraju Mar 25 '12 at 4:00