6.(a) Show that (f.g) = f f(x)g(x)dx satisfies the axioms of an inner product on C[a, b], the space of continuous functions on the interval [a, b]. (b) Define f(x) = cos nx and gn(x) = sin nx for n = 1,2,3,... Show that in the space C[0,27], we have that (fn. fm)= 0 and (gn. 9m) = 0 whenever m # n and that (fn. 9m) = 0 vm, n Show also that fall = √√= ||gn|| vn E N. (Thus, the set of vectors {n EN} is a set of mutually orthogonal unit vectors in C[a, b]. It is an orthonormal basis for a certain space (called a Hilbert Space) that is important in Fourier Analysis).
6.(a) Show that (f.g) = f f(x)g(x)dx satisfies the axioms of an inner product on C[a, b], the space of continuous functions on the interval [a, b]. (b) Define f(x) = cos nx and gn(x) = sin nx for n = 1,2,3,... Show that in the space C[0,27], we have that (fn. fm)= 0 and (gn. 9m) = 0 whenever m # n and that (fn. 9m) = 0 vm, n Show also that fall = √√= ||gn|| vn E N. (Thus, the set of vectors {n EN} is a set of mutually orthogonal unit vectors in C[a, b]. It is an orthonormal basis for a certain space (called a Hilbert Space) that is important in Fourier Analysis).
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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![6. (a) Show that (f. g) = f f(x) g(x)dx satisfies the axioms of an inner product
on C[a, b], the space of continuous functions on the interval [a, b].
(b) Define f(x) = cos nx and gn(x) = sin nx for n = 1,2,3,...
Show that in the space C[0,2π], we have that
(fnfm) = 0 and (gn. 9m) = 0 whenever mn and that (fn 9m) = 0 vm, n
Show also that ||fn|| = √√= ||gn|| Vn E N.
(Thus, the set of vectors {n E N} is a set of mutually orthogonal unit
vectors in C[a, b]. It is an orthonormal basis for a certain space (called a Hilbert
Space) that is important in Fourier Analysis).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f4a3967-3003-4c89-858c-e494ea95ddce%2F1a390b9b-639d-466e-b863-ccf1f326427f%2Fgr0mc3b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. (a) Show that (f. g) = f f(x) g(x)dx satisfies the axioms of an inner product
on C[a, b], the space of continuous functions on the interval [a, b].
(b) Define f(x) = cos nx and gn(x) = sin nx for n = 1,2,3,...
Show that in the space C[0,2π], we have that
(fnfm) = 0 and (gn. 9m) = 0 whenever mn and that (fn 9m) = 0 vm, n
Show also that ||fn|| = √√= ||gn|| Vn E N.
(Thus, the set of vectors {n E N} is a set of mutually orthogonal unit
vectors in C[a, b]. It is an orthonormal basis for a certain space (called a Hilbert
Space) that is important in Fourier Analysis).
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