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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

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 9E: Let D be an integral domain with four elements, D=0,e,a,b, where e is the unity. a. Prove that D has...
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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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