Equivalent definitions of an orthonormal function

Question

I want to prove that the following two definitions for an orthonormal function $\phi$, in terms of $kT$ time shifts, are equivalent.

So let $T$ the symbol period and $k$ an integer.

Definition 1

$ \int_{-\infty}^{\infty}\phi(t)\phi(t-kT)dt= \left\{ \begin{array}{ll} 1, & if\ k=0 \\ 0, & if\ k\neq 0 \\ \end{array} \right. $

Definition 2

$ \int_{-\infty}^{\infty}\phi(t-iT)\phi(t-jT)dt= \left\{ \begin{array}{ll} 1, & if\ i=j \\ 0, & if\ i\neq j \\ \end{array} \right. $

It looks pretty obvious but what could be a proof for this? Setting $k=i-j$ does not seem to produce an equivalent form.

Answer 1

Just do a substitution $t = t' + iT$: $$\int_{-\infty}^{\infty}\phi(t-iT)\phi(t-jT)\,\mathrm dt = \int_{-\infty}^{\infty}\phi(t'+iT-iT)\phi(t'+iT-jT)\,\mathrm dt' = \int_{-\infty}^{\infty}\phi(t')\phi(t'-(j-i)T)\,\mathrm dt'$$ Now set $k=j-i$ and rename $t'$ back to $t$.