Hello! Let us write such a code in WM9:

d = With[{n = 10}, ConstantArray[1, 2 n + 1] + ArrayPad[{1}, n]]
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

w = DiscreteWaveletTransform[d, MeyerWavelet[3]];

d2 = InverseWaveletTransform[w]
{1.12071802, 0.970766057, 0.95971983, 0.987282619, 1.08982838, \
0.943552369, 0.946829177, 1.04316409, 1.16005865, 0.930360322, \
1.86004042, 0.950989804, 1.09596701, 0.93382494, 0.956111119, \
1.06803454, 1.15908982, 0.963317658, 0.955032803, 0.989758086, \
1.05690014}

Possibly, it is a stupid question, but I cannot understand why InverseWaveletTransform does not return the initial series of data. It takes place only for MeyerWavelet. For any other wavelet I tried everything is OK/ E.g.:

  w2 = DiscreteWaveletTransform[d, DaubechiesWavelet[2]];

  InverseWaveletTransform[w2]
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}