Okay,

Ive done the first one,

LHS = sum ( int(f(x+n) x = 0 to 1), n = -inf, ..., inf)

consider the integral component,

inf(f(x+n) , x = 0 to 1)

here let t = x + n --> dt/dx = 1 or dt = dx

Change Limits: at x = 0 t = n, at x = 1 t = n +1

Thus,

inf(f(x+n), x = 0 to 1) becomes int( f(t) t = n to n+1) or int(f(x), x = n to n+1)

Plug back into the sum,

sum ( int(f(x), x = n to n + 1) , n = -inf,...,inf)

which as Im sure you can easily see is

sum(int(f(x), x = -inf...inf))

Hope this helps,

I'll keep plugging away on the rest,

Regards,

David