Saw tooth waves as basis to expand function?

[andy everett]

If a saw tooth wave can be represented by an alternating sum of sin
waves can a sin wave be represented by some sum of saw tooth waves?

Thank you for your help!

[Lee Rudolph]

andy everett <vze2...@verizon.net> writes:

>If a saw tooth wave can be represented by an alternating sum of sin
>waves can a sin wave be represented by some sum of saw tooth waves?

Yes (in the same sense of "represented by" and "sum").

Lee Rudolph

[Duder]

"andy everett" <vze2...@verizon.net> wrote in message
news:YzQxh.3550$xu4.2318@trndny04...

> If a saw tooth wave can be represented by an alternating sum of sin waves
> can a sin wave be represented by some sum of saw tooth waves?
>
> Thank you for your help!

no

saw tooth has harmonics of the fundamental frequency.

a sine wave does not.

a sin wave is only a fundamental frequency.

[Lee Rudolph]

"Duder" <nos...@nospam.com> writes:

That sequence of statements lacks something in explanatory power.

Lee Rudolph

[andy everett]

Lee Rudolph wrote:

Now that I know the answer is yes it is easy to sketch the alternating
sum of the first couple of "triangle" waves with wavelength L, L/3, L/5,
... such a sum starts to "get smooth"

Thank you!

[Duder]

"andy everett" <vze2...@verizon.net> wrote in message

news:f7Rxh.1058$Yl3.139@trndny09...

triangle wave = sum 1 to n [k*sin((2*n-1)*t)]

[david petry]

On Feb 5, 5:10 pm, andy everett <vze2q...@verizon.net> wrote:
> If a saw tooth wave can be represented by an alternating sum of sin
> waves can a sin wave be represented by some sum of saw tooth waves?

If g(x) = sum (k=1..oo) f(kx)/k^s

then f(x) = sum (k=1..oo) mu(k) g(kx)/k^s

where mu(k) is the moebius function.

Apply that to sin(x)

I'm doing this from memory. I hope I haven't messed up.

[Jean-Marie Aubry]

andy everett a écrit :

> If a saw tooth wave can be represented by an alternating sum of sin
> waves can a sin wave be represented by some sum of saw tooth waves?
>
> Thank you for your help!

I believe this kind of problem was studied in:

On some infinite series involving arithmetical functions
H Davenport - QJ Math, 1937

Hope this helps,

[John Bailey]

On Tue, 06 Feb 2007 01:47:55 GMT, andy everett <vze2...@verizon.net>
wrote:

>Lee Rudolph wrote:
>
>Now that I know the answer is yes it is easy to sketch the alternating
>sum of the first couple of "triangle" waves with wavelength L, L/3, L/5,
>... such a sum starts to "get smooth"

Thanks for an amusing challenge. You seem to have found your answer
but it took me a while to re-orient myself to waveform harmonics to
see the answer. Its a lot like division. The first order sawtooth
has sub-harmonics of period 1/2, 1/3, 1/4 etc. Use a second sawtooth
of 1/2 amplitude to suppress the second harmonic. Keeping track of
the new harmonics added to the result of this subtraction, just keep
subtracting the next remaining subharmonic until your sum starts to
"get smooth"

references:
Dissecting classical waveforms:
http://lena.ucsd.edu/~msp/techniques/latest/book-html/node187.html

harmonics of a sawtooth: (Java)
http://www.earlevel.com/Digital%20Audio/harmonigraf.html