       NFourierTrigSeries and NFourierCoefficient

• To: mathgroup at smc.vnet.net
• Subject: [mg114356] NFourierTrigSeries and NFourierCoefficient
• From: zosi <zosi at to.infn.it>
• Date: Thu, 2 Dec 2010 05:38:06 -0500 (EST)

```Dear All,

and  NFourierCoefficient with V7 or V8.
(everything OK with V6, with adequate FourierParameters).

Suppose I have a very simple function, first defined in (-1,4)
in two ways: Case A), with /;
and Case B) with Piecewise
and then extended periodically.

Case A)    (* it does work *)
tmin = -1; tmax = 4;   (* the interval is asymmetric around origin *)
period = tmax - tmin;
g[t_] := (t - 1)^2  /; tmin <= t <= tmax
g[t_] := g[t - period] /; t > tmax
g[t_] := g[t + period] /; t < tmin
Plot[g[t], {t, tmin, 2 tmax + 1}]
Needs["FourierSeries`"]
b = Solve[(2 \[Pi] )/bb == period, bb][[1, 1, 2]]
a = 1;
sergNTrig = NFourierTrigSeries[g[t], t, 7,
FourierParameters -> {a,b}] // Simplify
Plot[{g[t], sergNTrig}, {t, tmin, 2 tmax + 1}]    (* good reconstruction *)

Case B)  (* it fails                             *)
(* I just use Piecewise instead of  :;  *)

tmin = -1; tmax = 4;
period = tmax - tmin;
h[t_] := Piecewise[{{(t - 1)^2, tmin <= t <= tmax}}]
h[t_] := h[t - period] /; t > tmax
h[t_] := h[t + period] /; t < tmin
Needs["FourierSeries`"]
b = Solve[(2 \[Pi] )/bb == period, bb][[1, 1, 2]]
a = 1;
serhNTrig = NFourierTrigSeries[h[t], t, 7,
FourierParameters -> {a, b}] // Simplify
Plot[{h[t], serhNTrig}, {t, tmin, 2 tmax + 1}]  (* reconstruction fails *)

*)
Finally, **in both cases** I am not able to find the
right parameters (i.e., coefmul and {a,b}) in, e.g.,

direct\$coeff = Table[coefmul*NFourierCoefficient[h[t], t, n,
FourierParameters -> {a,b}], {n, 0, 7}]

to obtain directly the Fourier coefficients.

Where am I wrong ?