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8. Show that an element x of an inner product space X cannot have "too many" Fourier coefficients (x, ek) which are “big"; here, (e) is a given orthonormal sequence; more precisely, show that the number nm of (x, ek) such that (x, e₁)|>1/m must satisfy nm

8. Show that an element x of an inner product space X cannot have "too many" Fourier coefficients (x, ek) which are “big"; here, (e) is a given orthonormal sequence; more precisely, show that the number nm of (x, ek) such that (x, e₁)|>1/m must satisfy nm

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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8. Show that an element x of an inner product space X cannot have "too many" Fourier coefficients (x, e) which are “big”; here, (e) is a given orthonormal sequence; more precisely, show that the number nm of (x, ek) such that [(x, ek)|>1/m must satisfy nm <m² ||x||².
Transcribed Image Text:8. Show that an element x of an inner product space X cannot have "too many" Fourier coefficients (x, e) which are “big”; here, (e) is a given orthonormal sequence; more precisely, show that the number nm of (x, ek) such that [(x, ek)|>1/m must satisfy nm <m² ||x||².
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