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a) Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function f(x) expanded in this basis as f(x) = En Cnon(x), has expansion coefficients given by = ($n|f). Cn = b) Now consider another function g(x) expanded in the above basis as g(x) = Endnon(x). Prove that (flg) = Σncdn. c) Now consider a non-orthonormal basis {x} for this space, such that (XmIXn) = Smn. Gen- eralise the previous result to find an expression for (flg) in terms of Snm. Confirm that if Smn = 8mn, then this result reduces to the above one.

a) Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function f(x) expanded in this basis as f(x) = En Cnon(x), has expansion coefficients given by = ($n|f). Cn = b) Now consider another function g(x) expanded in the above basis as g(x) = Endnon(x). Prove that (flg) = Σncdn. c) Now consider a non-orthonormal basis {x} for this space, such that (XmIXn) = Smn. Gen- eralise the previous result to find an expression for (flg) in terms of Snm. Confirm that if Smn = 8mn, then this result reduces to the above one.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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a) Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function f(x) expanded in this basis as f(x) = En Cnon(x), has expansion coefficients given by = (onlf). Cn = b) Now consider another function g(x) expanded in the above basis as g(x) = Σn dnon (1). Prove that (flg) = Σncdn. c) Now consider a non-orthonormal basis {x} for this space, such that (XmIXn) = Smn. Gen- eralise the previous result to find an expression for (flg) in terms of Snm. Confirm that if Smn = 6mn, then this result reduces to the above one.
Transcribed Image Text:a) Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function f(x) expanded in this basis as f(x) = En Cnon(x), has expansion coefficients given by = (onlf). Cn = b) Now consider another function g(x) expanded in the above basis as g(x) = Σn dnon (1). Prove that (flg) = Σncdn. c) Now consider a non-orthonormal basis {x} for this space, such that (XmIXn) = Smn. Gen- eralise the previous result to find an expression for (flg) in terms of Snm. Confirm that if Smn = 6mn, then this result reduces to the above one.
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