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week14 - 14 Fourier analysis Function spaces Read Boas Ch 7...

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14 Fourier analysis Read: Boas Ch. 7. 14.1 Function spaces A function can be thought of as an element of a kind of vector space. After all, a function f ( x ) is merely a set of numbers, one for each point x of the underlying space. We can add functions in this way, componentwise, like vectors h ( x ) = f ( x )+ g ( x ), and (we will show below), we can define a metric, or distance function, on the set of all functions as well. It’s simplest to think about 1D first, a finite interval 0 x 2 L , and imagine “discretizing” this space so the N points in it are separated, like the gradations on a ruler by an amount Δ 2 L/N . A “vector” in function space | f i is therefore defined to be the set of components f 1 , f 2 , . . . f N representing the values of the function f at the points x 1 , x 2 , . . . . Now if we choose a basis of this space called a “position basis”, we define a vector | i i = 0 0 . . . 1 . . . 0 0 , (1) where the 1 is in the i th position, in other words the vector represents the position x i . This is clearly a basis for the vector space, since each vector is linearly inde- pendent and the whole space is spanned. The function f may now be represented as | f i = f 1 | 0 i + f 2 | 1 i + f 3 | 4 i · · · + f N | N i , (2) i.e. the function has the value f 1 f ( x 1 ) at x 1 , and so on. Note this space is finite-dimensional, but we can make L as large as we like, or choose N as large as we like..
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