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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 .

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    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 ...

    Solution Summary

    Properties of the Stone-Weierstrass Theorem are emphasized.

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