<p> Stem style construct </p>

c8940ea8-949a-4e95-861f-c8d17dd49097 36 6 12 decimal

&[DATE] &[TIME] - &[FILENAME]

Fourier integral wrt trig polynomials

L H N xd U 0 : dx H L - N / : i 1 N range U i el L dx - (dx i * + : for U 41 line :

L 1 :

N 128 :

Discontinuous function u f u L - ≤ 1 L - u ≤ (u 0 ≤ (& u - 0 u ≤ (u 1 ≤ (& 1 L u ≤ (u 3 L * ≤ (& u 2 L * - (3 L * u ≤ (u 4 L * ≤ (& 1 continue if if if if if :

X 2 L * - 4 L * N xd :

fnct i 1 N range u i el X i el : v i el X i el f : 21 line for u v augment 21 line :

fnct

.. -u ..

..(u-2*L)..

n 1 :

Project Notes:

u f u - :

1. Disontinuous function(s) must be defined piecewise, in oder of progression, completely. 2. wrt Fourier integral, Smath is not suited, not suited because Smath is not a "scalar CAS". 3. PROOF: coefficients [a, b] won't evaluate no matter their compute mode. In Smath, the definite integral is essentially "Simpson". 4. We will rescue the project via Budapest39. 5. At left, f(u):-u, f(u):(u-2*L), both are "scalar pieces".

1 L / u f n π * u * L / cos * u L - 0 int * 0.202642 -

u f u 2 L * - (:

1 L / u f n π * u * L / cos * u 0 2.5 int * 0.057834

x F 1 L / u f n π * u * L / cos * u L - x int * maple :

Discontinuous function u f u L - ≤ 1 L - u ≤ (u 0 ≤ (& u - 0 u ≤ (u 1 ≤ (& 1 L u ≤ (u 3 L * ≤ (& u 2 L * - (3 L * u ≤ (u 4 L * ≤ (& 1 continue if if if if if :

0.5 F 0.376144 -

a n 1 N range C n el 1 L / x f n π * x * L / cos * x L - L int * eval : for C 21 line :

b n 1 N range C n el 1 L / x f n π * x * L / sin * x L - L int * eval : for C 21 line :

ω 2 π N / * : 0.049087

Δx X N el X 1 el - (N 1 - / : 0.046875

u λ u X 1 el - (ω Δx / * :

0.05 λ 2.146755

Y fnct 2 col :

n 0 N range A n 1 + el 2 N / Y k 1 + el n k * ω * cos * k 0 N 1 - sum * eval : B n 1 + el 2 N / Y k 1 + el n k * ω * sin * k 0 N 1 - sum * eval : 21 line for

u B39 A 1 el 2 / A n 1 + el u λ n * cos * B n 1 + el u λ n * sin * + (n 1 N 2 / Floor sum 11 line + :

fit x 1.5 - 3.5 1.49 - range : i 1 x length range v i el x i el B39 eval : for x v augment 31 line :

Fourier Budapest39 Trig2discontinuous

fnct fit 2 1 sys

Q n 2 N range c n el A n el i (B n 1 - el * - : for c 21 line :

spectral t 1 N 2 range : i 2 N range v Q mat2sys abs sys2mat : v 1 el A 1 el (: 21 line for t v augment 31 line :

Isolated for stem style spec spectral 2 col :

DC 0 A 1 el 0022 mat :

<= Fake to view DC

<p> Stem style construct </p>

iii 1 N range q iii el 3 : q N el 2 : 21 line for

ii 1 N range v ii el ii : for

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

u q v replicate :

i 1 N range S 3 i * el spec i el : for

V u S augment :

V V 3 V rows 12 submatrix :

Unscaled spectrum. Remember: the 1rst component of a spectrum is the DC component, very small in this application [2*Mean(Y)]. The remaining of the spectrum components are quite large.

V DC 2 1 sys

Y Mean V Max 21 mat 0.592 12821 mat

A B augment