Fourier Transform 5 | Integrable Functions [dark version]

Q: Why does the integral-based quantity ∫_{−π}^{π} |f(x)| dx not automatically define a true norm on the raw set of functions?

Because it can be zero for nonzero functions. If f differs from the zero function only on points (or, more generally, on a set) that has no impact on the integral—such as a measure-zero set—then ∫_{−π}^{π} |f(x)| dx = 0 even though f is not identically zero. Norms must be positive definite, so this fails on the level of individual functions.

Q: How does the equivalence-class construction repair the norm problem in L1?

Two functions f and g are treated as equivalent when the integral of |f − g| over one period is zero. That means they can differ only on a set that integration cannot detect. The space is then built from equivalence classes, and the norm of a class is defined using any representative: ∥[f]∥_1 = ∫_{−π}^{π} |f(x)| dx. This is well-defined because equivalent representatives yield the same integral value.

Q: What changes when moving from L1 to L2?

L2 imposes a stronger integrability condition: ∫_{−π}^{π} |f(x)|^2 dx must be finite. This “square-integrable” requirement leads to a different norm (with a square root) and, importantly, it enables an inner product structure. That inner product is what makes orthogonal projections possible.

Q: Why is L2 the right setting for Fourier series via orthogonal projection?

Fourier series arises from projecting a function onto the subspace spanned by trigonometric polynomials. Orthogonal projection requires an inner product space. L2 is constructed so that the inner product and the associated geometry are valid, turning “best approximation by trigonometric polynomials” into a rigorous projection problem.

Q: How do trigonometric polynomials fit into these spaces?

Trigonometric polynomials form a subspace of the periodic function space, and they can be viewed inside L1 and L2 (up to the equivalence-class identification). In practical calculations, the equivalence-class step is often suppressed, so computations proceed as if working directly with functions, while the underlying theory treats them as classes when needed.

TL;DR

L1 for 2π-periodic functions consists of those with finite ∫_{−π}^{π} |f(x)| dx.

Briefing Cornell Notes

Briefing

Fourier analysis needs a precise function space where “integrable over one period” is enough to make approximation and projection work. For 2π-periodic complex-valued functions on the real line, that requirement leads to the space L1: functions whose absolute value has a finite integral over a single period, typically from −π to π. This is the threshold for treating trigonometric polynomials as a meaningful approximation target, but it also exposes a subtle problem—integrating |f| does not distinguish functions that differ only on sets the integral cannot “see.”

The construction starts with the vector space of 2π-periodic functions F: R → C and the trigonometric polynomials (finite linear combinations of sines and cosines, extended to complex coefficients). On the polynomial span, there is a natural inner product defined via an integral over one period. To broaden the setting beyond polynomials, the discussion moves to L1, defined by the finiteness of ∫_{−π}^{π} |f(x)| dx. While this gives a well-defined quantity, using it directly as a “norm” fails: the integral can be zero even when f is not the zero function—such as when f differs from zero only at jump points or, more generally, on sets of measure zero.

The fix is standard in analysis: replace individual functions with equivalence classes. Two functions f and g are declared equivalent if the integral of |f − g| is zero, meaning they can differ only on a set that has no impact under integration (measure-zero differences). The resulting quotient space—still denoted L1—becomes a genuine complex vector space where the norm is defined on equivalence classes by taking ∥[f]∥_1 = ∫_{−π}^{π} |f(x)| dx. Because equivalent representatives agree “almost everywhere,” this norm is positive definite and satisfies the usual norm axioms.

With L1 in place, the next step is strengthening integrability to support an inner-product structure. That leads to L2, the space of 2π-periodic functions for which ∫_{−π}^{π} |f(x)|^2 dx is finite. The L2 norm is defined using the square root of that integral, and—crucially—this stronger condition allows the norm to be tied to an inner product. Once L2 carries an inner product, orthogonal projections become available, which is exactly the mechanism behind Fourier series: projecting onto the subspace spanned by trigonometric polynomials yields the Fourier series approximation.

In short, the path runs from integrability (L1) to a quotient construction that removes measure-zero ambiguity, then to square-integrability (L2) to recover an inner product. That inner product is the mathematical engine that makes Fourier series approximations rigorous via orthogonal projection in the next installment.

Cornell Notes

Fourier approximation requires working in function spaces where integrals are finite and the geometry (inner products, projections) is well-defined. For 2π-periodic complex-valued functions, L1 consists of those with ∫_{−π}^{π} |f(x)| dx < ∞, but the naive “integral norm” fails to be positive definite because functions can differ on sets that integration cannot detect. The remedy is to identify functions that differ only on measure-zero sets, forming equivalence classes; the resulting quotient space L1 has a proper norm. To enable orthogonal projection and Fourier series, the discussion then moves to L2, where ∫_{−π}^{π} |f(x)|^2 dx < ∞. L2 supports an inner product, making projection onto trigonometric polynomials mathematically precise.

Why does the integral-based quantity ∫_{−π}^{π} |f(x)| dx not automatically define a true norm on the raw set of functions?

Because it can be zero for nonzero functions. If f differs from the zero function only on points (or, more generally, on a set) that has no impact on the integral—such as a measure-zero set—then ∫_{−π}^{π} |f(x)| dx = 0 even though f is not identically zero. Norms must be positive definite, so this fails on the level of individual functions.

How does the equivalence-class construction repair the norm problem in L1?

Two functions f and g are treated as equivalent when the integral of |f − g| over one period is zero. That means they can differ only on a set that integration cannot detect. The space is then built from equivalence classes, and the norm of a class is defined using any representative: ∥[f]∥_1 = ∫_{−π}^{π} |f(x)| dx. This is well-defined because equivalent representatives yield the same integral value.

What changes when moving from L1 to L2?

L2 imposes a stronger integrability condition: ∫_{−π}^{π} |f(x)|^2 dx must be finite. This “square-integrable” requirement leads to a different norm (with a square root) and, importantly, it enables an inner product structure. That inner product is what makes orthogonal projections possible.

Why is L2 the right setting for Fourier series via orthogonal projection?

Fourier series arises from projecting a function onto the subspace spanned by trigonometric polynomials. Orthogonal projection requires an inner product space. L2 is constructed so that the inner product and the associated geometry are valid, turning “best approximation by trigonometric polynomials” into a rigorous projection problem.

How do trigonometric polynomials fit into these spaces?