Integral of a Wavelet function

Question

I ran into problems integrating a function which contains both an If condition and a Wavelet which is an InterpolatingFunction knowing that the values of Phi are very small

ϕ:= WaveletPhi[DaubechiesWavelet[8],x]:
Dϕ:=Dt[ϕ[t],{t,1}];
derPhi[x_?NumericQ]:=If[x>= 0 && x<= 15,Dϕ[x],0];
NIntegrate[derPhi[x],{x,0.25,0.5}];

Answer 1

You introduce an unnecessary dummy variable x. This is my version

ϕ := WaveletPhi[DaubechiesWavelet[8]];
Dϕ := Derivative[1][ϕ];
derPhi[x_?NumericQ] := If[x >= 0 && x <= 15, Dϕ[x], 0];
NIntegrate[derPhi[x], {x, 0.25, 0.5}]

(* 0.00972165 *)

The numerical integration warns about slow convergence, presumably this is because of its many small discontinuities.

Is there a reason why you are differentiating and then integrating? Can't you just work with the original function?

ϕ[0.5] - ϕ[0.25]
(* 0.00972164 *)

Answer 2

The approach taken in this answer is also applicable to this problem.

Construct the function to be integrated:

derPhi = Head[Simplify[D[WaveletPhi[DaubechiesWavelet[8], x], x], 0 < x < 15]]

Verify that we have a piecewise linear interpolant:

derPhi["InterpolationOrder"]
   {1}

This means, we can use a low-order quadrature rule to evaluate (sub-)integrals between the interpolation points.

Verify how many interpolation points are used in the integration interval of interest:

Count[Flatten[derPhi["Grid"]], x_ /; 1/4 < x < 1/2]
   63

We can then use the "InterpolationPointsSubdivision" method of NIntegrate[]. The count above is well within the default setting of "MaxSubregions", so that doesn't need to be adjusted:

NIntegrate[derPhi[x], {x, 1/4, 1/2}, 
           Method -> {"InterpolationPointsSubdivision", 
                      Method -> {"GaussKronrodRule", "Points" -> 2}, 
                      "SymbolicProcessing" -> 0}]
   0.009721637451048174