Riemann–Lebesgue lemma

The lemma of Riemann - Lebesgue, also theorem of Riemann - Lebesgue, is a named after Bernhard Riemann and Henri Lebesgue mathematical theorem from calculus. He states that the Fourier transforms of integrable functions vanish at infinity.

Wording of the sentence

It is an integrable function, that is

Is the Fourier transform

So true, ie vanishes at infinity.

Specifically, the statement that there exists for every real number a such that for all.

Since the Fourier transforms of integrable functions are continuous, it is in a continuous function which vanishes at infinity, such functions are called C0 - functions. Therefore, one can also formulate the lemma of Riemann - Lebesgue briefly as: Fourier transform of integrable functions are C0 functions.

Evidence

The very simple proof to be presented here in outline. For returns the substitution

And we have a second formula for. If we now form the average of the two formulas and takes sums it pulls under the integral, making the exponential term to 1, it follows

And converges to the set of the dominated convergence for 0 against

Other evidence shows the first set of smooth functions with compact support and then exploit the fact that these are dense in the space of integrable functions.

Generalizations

Functions of several variables

The lemma of Riemann - Lebesgue can be generalized to functions:

It is an integrable function, that is

Is the Fourier transform

Shall be applied for.

It is standard on any of the, for example, the Euclidean norm.

Banach algebras

The amount of the integrable functions, that is, the amount of L1 functions, together with the folding of the multiplication, and the 1- norm, a Banach. In the harmonic analysis, it shows that the Fourier transformation is a special case of the abstract Gelfand transformation. The lemma of Riemann - Lebesgue then follows from the fact that the Gelfand transform in the space of C0 functions maps and the Gelfand space can be identified by with. At the same time by the lemma of Riemann - Lebesgue is generalized to locally compact abelian groups.