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posted Thursday April 26
In class I'll demonstrate some graphs of
partial sums of Fourier Series using a Function grapher at
http://www.flashandmath.com/mathlets/calc/funseq/funseq.html.
We will view their Example of Series 4 and
another example.
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posted Thursday April 19
In class today we talked about dense subsets
of L_p in Fact 1.4.
You can read up on Fact 1.4 in Werner's book
Einführung in die
höhere Analysis.
In the first edition, see Satz IV.7.6 (which uses Satz IV.4.6), Satz IV.7.7, and Korollar IV.7.8. Note in his Satz IV.4.6, he divided the
domain into interval of lengh 1/2^n where is class we used 1/n;
this way he gets monotone functions in part (a).
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posted Thursday April 19
In class today we reviewed the basics on Hilbert spaces.
You can read up on this in Werner's book
Einführung in die
höhere Analysis.
Specifically, taking the numbers from the first edition:
- our Def 1.5 is his Def. V.3.1, Lemma V.3.3, and Def. V.3.4
- our Example 1.6 is his Beispiele on Seite 317 (b) and (c)
- our Fact 1.8 is his Def V.4.1 and Satz V.4.8.
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posted Wednesday April 18
On a question after class over a theorem by Lebesgue:
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Let f:[a,b] --> R be a function.
Then [f is Riemann integrable] if and only if
[the set of discontinuities of f is a set of
Lebesgue measure zero
and f is bounded].
- Proof one: see Trench Theorem 3.5.6.
- Proof two:
Prof.
Schep, which is published in Stoll's book on Real Analysis.
Bottom line: a Riemann Integrable function f:[a,b] --> R is necessarily
a bounded function, no matter how you define the Riemann integral.
So the function f:[0,1] --> R defined by
f(x) = 1/x if x is in (0,1] and f(0)=17 is not Riemann integrable,
although it's set of discontinuity is a set of measure zero.
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There are many ways to define the Riemann Integral.
Here are 4 equivalent definitions, which are
notes from
Trench, Real Analysis book.
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The first definition of the notes is Trench's Definition 3.1.1.
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The third definition of the notes is Trench's Definition 3.1.3.
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The first and second definition of these notes can be defined for
any function f:[a,b] --> R. Then it can be shown that
such a function satisfying this condition must be bounded
(see Trench Theorem 3.1.2).
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The third and fourth definition of these notes only make sense if
the function f:[a,b] --> R is assumed to be bounded from the start
since you need to take the max and min of
the function on a subinterval
(thus we assume f is bounded).
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That definition (1) and (3) are equivalent for bounded functions,
see Trench Theorem 3.2.6.
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That definition (1) and (4) are equivalent for bounded functins,
see Trench Theorem 3.2.7.
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