Orthogonal Function Systems

Fourier Analysis · Axiom Academy

LESSON Orthogonal Function Systems Generalizing Fourier Analysis Beyond Sines and Cosines Just as vectors have dot products, functions have inner products . For functions f and g on an interval [a,b] with a weight function w(x), the inner product is: The weight function w(x) > 0 allows us to emphasize certain parts of the interval. Different weight functions lead to different orthogonal systems. Two functions φ m and φ n are orthogonal with respect to weight w(x) if their inner product is zero when m ≠ n: A family φ n is an orthogonal system if every pair of distinct functions is orthogonal. If additionally ||φ n || = 1, the system is orthonormal . 3. Classical Orthogonal Polynomials Different weight functions and intervals produce different orthogonal polynomial families, each with unique properties: 4. Generalized Fourier Coefficients Just as we expand functions in terms of sines and cosines, we can expand any function f(x) in terms of an orthogonal system φ n : The coefficients c n are found using the orthogonality property:

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