# A functional that returns the function's value outside integration bounds

We have a functional over $$\mathbb {R} \to \mathbb {R}$$ functions $$\{f\}$$, that could be written as

$$F (f) := \int_{-\infty}^\infty \mathbb{d}t K (t) f (t)$$,

where $$K(t)$$ is a distribution that could be a function or a generalized function (such as Dirac delta).

If we wish for the functional to extract the functions's value at $$t_0$$, we simply take $$K(t) = \delta (t - t_0)$$.

However, what if we concentrate solely on a subsection of the integration space, i.e. lets define the functional as

$$F_+ (f) := \int_{0}^\infty \mathbb{d}t K (t) f (t)$$, ($$f$$ is still well defined on whole $$\mathbb{R}$$),

we now wish to find again $$K (t)$$ such that $$F_+ (f) = f (t_0)$$, for $$t_0 > 0$$ this is again trivially the Dirac delta, but what about $$t_0 < 0$$? Is there any general distribution $$K (t)$$ such that the distribution would give $$f (t_0), t_0 < 0$$, for an arbitrary function?

If not, is there $$K (t)$$ with such properties of a specific class of functions?

I.e. if we have a class of functions

$$g_{a, b} (t) := \frac {a t} {(a b)^2 + (t - b)^2}$$, $$\mathbb {R} \to $$, $$a, b > 0$$,

characterised by 2 positive real parameters, does there exist $$K (t)$$, possibly dependent on $$a$$, neccesarily indepentent of $$b$$, such that

$$F_+ (g_{a, b}) := \int_{0}^\infty \mathbb{d}t K_a (t) g_{a, b} (t) = g_{a, b} (t_0), t_0 < 0$$?

• There definitely won't be a general formula, since a general function's value at $t=-1$ (say) is completely independent of its values for $t\ge0$. Mar 18 at 20:18

I'll cheat a little bit : if you allow $$K(t)$$ to be an operator, as $$K(t) = \delta(t-\tau)\,e^{(t_0-\tau)\partial_t}$$ for instance, with $$\tau > 0$$ and $$t_0 < 0$$, then you can construct some relations as follows : $$F[f] = \int_0^\infty K(t)f(t) \,\mathrm{d}t = \int_0^\infty \delta(t-\tau)\,e^{(t_0-\tau)\partial_t}f(t) \,\mathrm{d}t = \int_0^\infty \delta(t-\tau)f(t+t_0-\tau) \,\mathrm{d}t = f(t_0)$$