Return to Question

Definition $\mathcal{M}_a$, the space of absolutely null moment functions, is the set of all measurable $f$ satisfying $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots. $$ $\mathcal{M}$, the space of null moment functions, is the set of all measurable $f$ satisfying $$ \int_0^T |f(t)| dt < \infty \quad \forall \\, T>0, \qquad \lim_{R \to \infty} \int_0^R t^n f(t) dt = 0 \quad \forall \\, n=0,1,2,\ldots $$

###Original QUESTION 1### If $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots, $$ when must $f \equiv 0$ almost everywhere?

Example 2: by taking imaginary parts of $\int_0^\infty t^{4n+3} \exp(-(1+i)t) dt \in \mathbb{R}$, we get $$ \int_0^\infty t^{4n+3} e^{-t} \sin t \\, dt = 0, $$ and so by substituting $t = x^{1/4}$, $$ \int_0^\infty x^n \exp(-x^{1/4}) \sin(x^{1/4}) dx = 0, \qquad n=0,1,2,\ldots $$

Definition $\mathcal{M}_a$, the space of absolutely null moment functions, is the set of all measurable $f$ satisfying $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \, n=0,1,2,\ldots. $$ $\mathcal{M}$, the space of null moment functions, is the set of all measurable $f$ satisfying $$ \int_0^T |f(t)| dt < \infty \quad \forall \, T>0, \qquad \lim_{R \to \infty} \int_0^R t^n f(t) dt = 0, \quad \forall \, n=0,1,2,\ldots $$

###Original QUESTION 1### If $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \, n=0,1,2,\ldots, $$ when must $f \equiv 0$ almost everywhere?

Example 2: by taking imaginary parts of $\int_0^\infty t^{4n+3} \exp(-(1+i)t) dt \in \mathbb{R}$, we get $$ \int_0^\infty t^{4n+3} e^{-t} \sin t \, dt = 0, $$ and so by substituting $t = x^{1/4}$, $$ \int_0^\infty x^n \exp(-x^{1/4}) \sin(x^{1/4}) dx = 0, \qquad n=0,1,2,\ldots $$

Hi, apologies if these questions are trivial and/or stupid.

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple here).

QUESTION 1 (surely must be known?) If $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots, $$ when must $f \equiv 0$ almost everywhere?  EDIT: CLARIFICATION: this is really about classes of functions, expressed in terms of a growth/decay rate function $\phi$, which give unique solutions to the moment problem. I am NOT asking how to solve the moment problem itself!

QUESTION 2 (probably much harder; maybe unknown?) If instead we have only $\int_0^R |f(t)| dt < \infty$ for each $R>0$, and $$ \lim_{R \to \infty} \int_0^R t^n f(t) dt = 0, \qquad n=0,1,2,\ldots $$ when must $f \equiv 0$ almost everywhere? (I have very little idea about this).

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple here).

###FINAL CLARIFICATION/REWRITE!!### Thanks to all who have commented so far. I will need some more time to digest it properly. The original forms of the questions are at the end; here I have rewritten the questions, hopefully more clearly; sorry for my poor explanation before!

Definition $\mathcal{M}_a$, the space of absolutely null moment functions, is the set of all measurable $f$ satisfying $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots. $$ $\mathcal{M}$, the space of null moment functions, is the set of all measurable $f$ satisfying $$ \int_0^T |f(t)| dt < \infty \quad \forall \\, T>0, \qquad \lim_{R \to \infty} \int_0^R t^n f(t) dt = 0 \quad \forall \\, n=0,1,2,\ldots $$

Thus, trivially $0 \in \mathcal{M}_a \subseteq \mathcal{M}$, but $\mathcal{M}_a$ contains many other non-trivial functions. It seems certain that $\mathcal{M}_a \ne \mathcal{M}$ (I would be amazed if the spaces were equal), although constructing an explicit example seems tricky.

Definition Given a function $\psi \geq 0$, let $G(\psi)$ be the set of all $f$ such that $|f| \leq \psi$.

(Of course we're identifying functions equal a.e., so really we should consider equivalence classes etc. just as for $L^p$ spaces).

Rephrased Question 1 Find general simple necessary and/or sufficient conditions on $\psi$ with the property $$ G(\psi) \cap \mathcal{M}_a = \{ 0 \}. $$

Rephrased Question 2 The same as Question 1, but with $G(\psi) \cap \mathcal{M}$ instead.

Or, if $G(\psi)$ is not the appropriate space for these problems, consider $\int_T^{2T} |f| dt \leq g(T) $ or $\int_n^{n+1} |f| dt \leq A_n$ or something similar instead. Finding the correct kind of restrictions on $f$ is part of the problem.

Thus, $G(\psi) \cap \mathcal{M}_a = \{ 0 \}$ for $\psi(t) = \exp(-\delta t)$, by the discussion below; and also for any compactly supported $\psi \in L^1$.

But $G(\psi) \cap \mathcal{M}_a \ne \{ 0 \}$ for $\psi(t) = \exp(-t^{1/4}) |\sin(t^{1/4})|$.

###Original QUESTION 1### If $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots, $$ when must $f \equiv 0$ almost everywhere? 

EDIT: CLARIFICATION: this is really about classes of functions, expressed in terms of a growth/decay rate function $\phi$, which give unique solutions to the moment problem. I am NOT asking how to solve the moment problem itself!

###Original QUESTION 2### If instead we have only $\int_0^R |f(t)| dt < \infty$ for each $R>0$, and $$ \lim_{R \to \infty} \int_0^R t^n f(t) dt = 0, \qquad n=0,1,2,\ldots $$ when must $f \equiv 0$ almost everywhere? (I have very little idea about this).

QUESTION 1 (surely must be known?) If $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots, $$ when must $f \equiv 0$ almost everywhere? EDIT: CLARIFICATION (since several people seemed to misinterpret my question): this is really about classes of functions, expressed in terms of a growth/decay rate function $\phi$, which give unique solutions to the moment problem. I am NOT asking how to solve the moment problem itself!

QUESTION 1 (surely must be known?) If $$ \int_0^\infty t^n |f(t)| dt < \infty, \quad \int_0^\infty t^n f(t) dt = 0, \qquad \forall \\, n=0,1,2,\ldots, $$ when must $f \equiv 0$ almost everywhere? EDIT: CLARIFICATION: this is really about classes of functions, expressed in terms of a growth/decay rate function $\phi$, which give unique solutions to the moment problem. I am NOT asking how to solve the moment problem itself!