Fourier Transform 5 | Integrable Functions

Q: Why does the integral of |f(x)| over a period not automatically define a valid norm on the space of 2π-periodic functions?

Because it can be zero for a function that is not identically zero. If two functions differ only on points (or more generally on a set of measure zero), the integral of |f(x)| can’t “see” those differences. That means a nonzero function can still have ∫_{-π}^{π} |f(x)| dx = 0, violating positive definiteness, which a norm requires.

Q: What equivalence relation fixes the norm problem in L1?

Two functions f and g are treated as equivalent when ∫_{-π}^{π} |f(x) − g(x)| dx = 0. This captures the idea that f and g may differ only on a set of measure zero. The L1 space is then defined as the set of these equivalence classes (a quotient space), not the raw functions themselves.

Q: How is the L1 norm defined after passing to equivalence classes?

For an equivalence class [f], the norm is defined as ∥[f]∥ = ∫_{-π}^{π} |f(x)| dx (using any representative f). It is well-defined because any other representative g in the same class differs from f only on a measure-zero set, so the integral of |g| matches the integral of |f|.

Q: What changes when moving from L1 to L2?

L2 strengthens the integrability requirement: instead of ∫ |f(x)| dx being finite, it requires ∫ |f(x)|^2 dx < ∞. This produces a different norm (the square root of that integral) and, more importantly, an inner product structure. That inner product enables orthogonal projections, which are the geometric mechanism behind Fourier series.

Q: Why does L2 matter for Fourier series specifically?

Fourier series are built using orthogonal projection onto the subspace spanned by trigonometric polynomials. Orthogonal projection requires an inner product, and the transcript indicates that L2 is the space where the inner product and associated orthogonality concepts are available in a way compatible with the integral-based definitions.

TL;DR

L1 for 2π-periodic functions requires finiteness of ∫_{-π}^{π} |f(x)| dx.

Briefing Cornell Notes

Briefing

Fourier analysis for 2π-periodic functions hinges on choosing the right function spaces—especially the integrability conditions that make inner products and norms mathematically well-behaved. The starting point is the set of 2π-periodic functions from real numbers to complex numbers, the same class used to approximate signals with cosine and sine. Trigonometric polynomials form a subspace built from these periodic building blocks, and on that subspace an inner product is defined using an integral over one period. To extend that machinery beyond polynomials, the framework needs a space where the integral operations used in Fourier coefficients actually converge.

That requirement leads to the integrable-function space L1. A 2π-periodic function belongs to L1 when its absolute value is integrable over a full period—specifically, when ∫_{-π}^{π} |f(x)| dx is finite. While it can be treated like an ordinary Riemann integral for calculation, the theory relies on Lebesgue integration to support the functional-analytic structure. With addition and scalar multiplication inherited from ordinary functions, L1 becomes a vector space once integrability is guaranteed.

However, using the raw quantity ∫ |f| as a “norm” runs into a problem: it is not positive definite. Functions can differ on sets that are invisible to the integral—such as changes at isolated points—yet still have the same integral of absolute value. In that situation, a nonzero function can end up with “norm zero,” which violates the norm axiom needed for a proper metric structure.

The standard fix is to stop treating functions as literal objects and instead treat them as equivalence classes. Two functions f and g are declared equivalent when the integral of |f(x) − g(x)| over one period is zero, meaning they can differ only on a set of measure zero. The resulting quotient space is again denoted L1, but now it is a genuine complex vector space where the norm becomes positive definite: the norm of an equivalence class is defined using any representative function, and it is well-defined because all representatives agree in the integral sense.

With L1 in place, the next step strengthens integrability to square-integrability, producing L2. A function is in L2 when ∫_{-π}^{π} |f(x)|^2 dx is finite. This change yields a different norm (built from the square root of that integral) and, crucially, an inner product structure. Once L2 carries an inner product, orthogonal projections become available—exactly the tool needed to approximate a general 2π-periodic function by its trigonometric-polynomial component.

The payoff is clear: Fourier series can be framed as an orthogonal projection onto the subspace spanned by trigonometric polynomials, but that projection only makes sense after building L1 and L2 carefully so integrals, norms, and inner products behave properly. The next installment is set to use this inner-product geometry to construct the Fourier series via orthogonal projection.

Cornell Notes

The construction of Fourier series depends on working in function spaces where integrals define norms and inner products reliably. For 2π-periodic complex-valued functions, L1 requires ∫_{-π}^{π} |f(x)| dx < ∞, but the integral-based quantity fails to be positive definite because functions can differ on measure-zero sets. The remedy is to identify functions that agree “almost everywhere,” forming equivalence classes; the quotient space (still called L1) then supports a true norm. To get an inner product and orthogonal projections, the framework moves to L2, where ∫_{-π}^{π} |f(x)|^2 dx < ∞. In L2, the norm comes from an inner product, setting up Fourier series as an orthogonal projection onto trigonometric polynomials.

Why does the integral of |f(x)| over a period not automatically define a valid norm on the space of 2π-periodic functions?

Because it can be zero for a function that is not identically zero. If two functions differ only on points (or more generally on a set of measure zero), the integral of |f(x)| can’t “see” those differences. That means a nonzero function can still have ∫_{-π}^{π} |f(x)| dx = 0, violating positive definiteness, which a norm requires.

What equivalence relation fixes the norm problem in L1?

Two functions f and g are treated as equivalent when ∫_{-π}^{π} |f(x) − g(x)| dx = 0. This captures the idea that f and g may differ only on a set of measure zero. The L1 space is then defined as the set of these equivalence classes (a quotient space), not the raw functions themselves.

How is the L1 norm defined after passing to equivalence classes?

For an equivalence class [f], the norm is defined as ∥[f]∥ = ∫_{-π}^{π} |f(x)| dx (using any representative f). It is well-defined because any other representative g in the same class differs from f only on a measure-zero set, so the integral of |g| matches the integral of |f|.

What changes when moving from L1 to L2?

L2 strengthens the integrability requirement: instead of ∫ |f(x)| dx being finite, it requires ∫ |f(x)|^2 dx < ∞. This produces a different norm (the square root of that integral) and, more importantly, an inner product structure. That inner product enables orthogonal projections, which are the geometric mechanism behind Fourier series.

Why does L2 matter for Fourier series specifically?