I am attempting an exercise in functional analysis: Let $PC_2[0,1]$ denote the vector space of piecewise continuous functions $f$ on $[0,1]$ such that $\int_0^1|f(t)|^2dt < \infty $. Then I can show that indeed $<f, g> =\int^1_0 f(t)\overline{g(t)} dt$ defines an inner product on $PC^2[0, 1]$. Now my difficulty is I want to:
a) Show the set of continuous functions in $PC^2[0, 1]$ is dense in $PC^2[0, 1]$. So given any piecewise continuous $f$, there's a continuous $g$ so that $\int (f - g)^2 < \epsilon$. Given such an $f$, there are at most a finite number $x_1 < ... <x_N$ points of discontinuity. So by set $g = f$ for the most of the intervals? I am not sure of the details hmm also, how to prove the same holds for any subset of step functions?
b) Show that 1 and the Haar functions form an othonormal basis for $PC^2[0, 1]$. From Haar system forms an orthonormal system in $L_2[0,1]$, we can show inductively that $$\chi_{[k2^{-n},(k+1)2^{-n}]}$$ lies in the span for all $0≤k<2^n$. This implies $\chi_{[a,b]}$ lies in the span (as an element of $L^2$, ie differing from that function on a null-set) whenever $a,b$ are of the form $2^n k$ for some $n,k$. That implies (since such points are dense in $[0,1]$) that the characteristic function of any sub-interval lies in the closure of the span, but the closure of that is $L^2[0,1]$ so the span of the wavelets was dense.
Start with $n=1$, $$\psi_{0,0}+1=2\chi_{[0,1/2]}\qquad 1-\psi_{0,0}=2\chi_{[1/2,1]}$$
Now $\psi_{n,k}=\chi_{[k2^{-n},(k+1/2)2^{-n}]}-\chi_{[(k+1/2)2^{-n},(k+1)2^{-n}]}$. From induction we are supposing $\chi_{[k2^{-n},(k+1)2^{-n}]}$ is in the span, so it follows that $$\chi_{[k2^{-n},(k+1/2)2^{-n}]}-\chi_{[(k+1/2)2^{-n},(k+1)2^{-n}]}+\chi_{[k2^{-n},(k+1)2^{-n}]}=2\chi_{[k2^{-n},(k+1/2)2^{-n}]}=2\chi_{[(2k)2^{-(n+1)},(2k+1)2^{-(n+1)}]}$$ Lies in the closure of the span for any $0≤2k<2^{n+1}$. The odd case follows by considering $$\chi_{[k2^{-n},(k+1)2^{-n}]}-(\chi_{[k2^{-n},(k+1/2)2^{-n}]}-\chi_{[(k+1/2)2^{-n},(k+1)2^{-n}]})=2\chi_{[(k+1/2)2^{-n},(k+1)2^{-n}]}=2\chi_{[(2k+1)2^{-(n+1)},(2k+2)2^{-(n+1)}]}$$. Does this solution work for this case?
c) Show the Rademacher functions form an orthonormal sequence in $PC^2[0, 1]$. From Rademacher functions form an orthonormal system but not an orthonormal basis, I see that Rademacher functions form an orthonormal system for $L_2[0,1]$: ie To compute $\langle r_n, r_m\rangle$ with $n<m$, say, observe that $$\langle r_n, r_m\rangle=\int_0^1 \operatorname{sgn}(\sin(2^n\pi t))\operatorname{sgn}(\sin(2^m\pi t))\,dt\\=\sum_{k=0}^{2^n-1}\int_{k2^{-n}}^{(k+1)2^{-n}} \operatorname{sgn}(\sin(2^n\pi t))\operatorname{sgn}(\sin(2^m\pi t))\,dt$$ In each summand, $\operatorname{sgn}(\sin(2^n\pi t))$ is constant and $\operatorname{sgn}(\sin(2^m\pi t))$ runs over $2^{m-n}$ full periods. Thus each summand is zero. Is this same proof correct for this case?
d) Show the Walsh function form an orthonormal basis for $PC^2[0, 1]$. From prove Walsh functions form a closed orthonormal system, there is a proof showing Walsh functions are orthogonal with respect to the $L_2$ inner product but can this be used to show orthonormality for $PC^2[0, 1]$?
Thank You for any help :)
