| Summary: | In this paper, we present and justify a methodology to solve the Monge–Kantorovich mass transfer problem through Haar multiresolution analysis and wavelet transform with the advantage of requiring a reduced number of operations to carry out. The methodology has the following steps. We apply wavelet analysis on a discretization of the cost function level <i>j</i> and obtain four components comprising one corresponding to a low-pass filter plus three from a high-pass filter. We obtain the solution corresponding to the low-pass component in level <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>μ</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow><mo>*</mo></msubsup></semantics></math></inline-formula>, and using the information of the high-pass filter components, we get a solution in level <i>j</i> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>μ</mi><mo stretchy="false">^</mo></mover><mi>j</mi></msub></semantics></math></inline-formula>. Finally, we make a local refinement of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>μ</mi><mo stretchy="false">^</mo></mover><mi>j</mi></msub></semantics></math></inline-formula> and obtain the final solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>μ</mi><mrow><mi>j</mi></mrow><mi>σ</mi></msubsup></semantics></math></inline-formula>.
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