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    Let be a compact interval and let A be a collection of continuous functions on which satisfy the properties of the Stone-Weierstrass Theorem
    [Stone-Weierstrass Theorem: Let K be a compact subset of and let A be a collection of continuous functions on K to R with the properties:
    a) The constant function belongs to A.
    b) If
    c) if
    d) if
    Then any continuous function K to R can be uniformly approximated on K by functions in A. ]

    Show that any continuous function on to R can be uniformly approximated by functions of the form where .

    © BrainMass Inc. brainmass.com December 24, 2021, 9:22 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/properties-stone-weierstrass-theorem-371064

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    Let be a compact interval and let A be a collection of continuous functions on which satisfy the properties of the Stone-Weierstrass Theorem
    [Stone-Weierstrass Theorem: Let K be a compact subset of and let A be a collection of continuous functions on K to R with the properties:
    a) The constant function belongs to A.
    b) If
    c) if
    d) if
    Then any continuous function K to R can be uniformly approximated on K by functions in A. ]

    Show that any continuous function on to R can be uniformly approximated by functions of the form where .
    -----------------
    Since is compact, we just need to show that the collection of functions generated by the set , that is, the collection of all linear conbinations of the form , satisfies the conditions a) - d)above.

    Show that if and are continuous at and respectively, then is also continuous at .

    First, notice, that if is continuous at , then the function is continuous at for any

    Indeed, is continuous at means that for every epsilon bigger than zero there's a delta bigger than zero such that if , then

    is continuous at means that for every epsilon bigger than zero there's a delta such that if then

    Given , find such that as soon as , we have
    Now, let be a point such that . Then we have:
    Since , we have , and so

    Similarly, we can show that if is continuous at , then is continuous at for any

    Now, represent and use the fact that the product of contunious function is continuous.

    Now, let's check the conditions a) - d) for the elements of :
    a) Since the constant function belongs to A, we have that the constant function belongs to .
    b) Since is defined as the set of all linear combinations of functions of the form
    , we have this condition fulfilled automatically.
    c) It's enough to check that if and ,
    then , and
    by the properties of we have that both and belong to .
    d) Given two points . We need to show that there's a function in such that .

    If then either or (or both).
    Suppose, .
    Since A satisfies the hypothesis of Stone-Weierstrass theorem, there's a function f in A such that . Consider the function where e(y) is the constant function . Then, .
    If then we must have . Find a function such that and define . We'll have .

    Thus, the conditions of Stone-Weierstrass Theorem are fulfilled for the collection , and we conclude that every fontinuous function can be uniformly approximated by the elements of , that is, the functions of the form where

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 9:22 pm ad1c9bdddf>
    https://brainmass.com/math/graphs-and-functions/properties-stone-weierstrass-theorem-371064

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