Eigenfunction Expansions

Fourier Analysis · Axiom Academy

LESSON Eigenfunction Expansions Representing functions as infinite series of orthogonal eigenfunctions Given a complete orthogonal set of eigenfunctions φₙ(x) , any "well-behaved" function f(x) can be expressed as an infinite series: The eigenfunctions φₙ(x) come from a Sturm-Liouville problem and satisfy orthogonality conditions. Each term in the series represents a "mode" or "component" of the function. To find the coefficient cₙ for each eigenfunction, we use the orthogonality property. Taking the inner product of both sides with φₙ(x): This formula is analogous to Fourier coefficients. The numerator measures "how much" of φₙ is in f, while the denominator normalizes by the "size" of φₙ. With weight function w(x), the inner product is: 3. Convergence and Completeness For the expansion to work, the eigenfunction set must be complete (span the entire function space). Under appropriate conditions: This convergence is in the L² sense (mean-square). The partial sum Sₙ approaches f as N → ∞. The error decreases according to Bessel's inequality and Parseval's identity: 4. Modal Analysis in Vibration Problems In physics and engineering, eigenfunction expansions describe vibrating systems. Each eigenfunction represents a mode shape , and each coefficient represents the mode's amplitude: For a vibrating string or beam, φₙ(x) is the nth mode shape, and the time evolution u(x,t) is given by superposition of modes. Each mode oscillates at its natural frequency ωₙ:

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