1.4 Convergence of Fourier series in L2 norm

Fourier series convergence in L2 norm is a key concept in harmonic analysis. It measures how well a function can be approximated by its Fourier series using the L2 norm, which captures the "average" difference between functions.

This idea connects to broader themes of function spaces, orthogonality, and approximation theory. Understanding L2 convergence helps us analyze how well Fourier series represent functions and their properties in various applications.

L2 Norm and Convergence

L2 Norm and Mean Square Convergence

• L2 norm measures the size of a function in the L2 space, defined as $\|f\|_2 = \left(\int_a^b |f(x)|^2 dx\right)^{1/2}$
• L2 norm is a way to quantify the distance between functions in the L2 space
• Mean square convergence refers to the convergence of a sequence of functions $\{f_n\}$ to a function $f$ in the L2 norm, i.e., $\lim_{n\to\infty} \|f_n - f\|_2 = 0$
• Mean square convergence implies pointwise convergence almost everywhere

Completeness and Hilbert Space

• Completeness is a property of a normed vector space where every Cauchy sequence converges to an element within the space
• L2 space is a complete normed vector space with respect to the L2 norm
• Hilbert space is a complete inner product space, which generalizes the notion of Euclidean space to infinite dimensions
• L2 space is an example of a Hilbert space, where the inner product is defined as $\langle f, g\rangle = \int_a^b f(x)\overline{g(x)} dx$

Inequalities and Identities

Bessel's Inequality and Parseval's Identity

• Bessel's inequality states that for an orthonormal system $\{\varphi_n\}$ in a Hilbert space and any element $f$ in the space, $\sum_{n=1}^\infty |\langle f, \varphi_n\rangle|^2 \leq \|f\|^2$
• Bessel's inequality provides an upper bound for the sum of the squares of the Fourier coefficients of a function
• Parseval's identity is a special case of Bessel's inequality when equality holds, i.e., $\sum_{n=1}^\infty |\langle f, \varphi_n\rangle|^2 = \|f\|^2$
• Parseval's identity states that the sum of the squares of the Fourier coefficients equals the energy (L2 norm squared) of the function
• Parseval's identity can be used to prove the completeness of the Fourier system in the L2 space

Riemann-Lebesgue Lemma

• Riemann-Lebesgue lemma states that for any integrable function $f$ on $[a, b]$, the Fourier coefficients $a_n$ and $b_n$ tend to zero as $n$ tends to infinity
• Riemann-Lebesgue lemma implies that the high-frequency components of a Fourier series have diminishing contributions to the function
• Riemann-Lebesgue lemma is a consequence of the properties of the Fourier transform and the integrability of the function
• Riemann-Lebesgue lemma is important in understanding the behavior of Fourier series and their convergence properties