Radialfunctionshavebecomeausefultoolinnumericalmathematics.Onthespheretheyhavetobeidentifiedwiththezonalfunctions.Weinvestigatezonalpolynomialswithmassconcentrationatthepole,inthesenseoftheirL1-normisattainingtheminimumvalue.Suchpolynomialssatisfyacomplicatedsystemofnonlineare-quations(algebraicifthespacedimensionisodd,only)andalsoasingulardifferentialequationofthirdorder.Theexactorderofdecayoftheminimumvaluewithrespecttothepolynomialdegreeisdetermined.Byourresultswecanprovethatsomenodalsystemsonthesphere,whicharedefinedbyaminimum-property,areprovidingfundamentalmatriceswhicharediagonal-dominantorboundedwithrespecttothe∞-norm,atleast,asthepolynomialdegreetendstoinfinity.
