InthispaperweconsiderpolynomialsplinesS(x)withequidistantnodeswhichmaygrowa5O(|x|~5).Wepresentanintegralrepresentationofsuchsplineswithadistributionkernel.Thisrepre-sentationisrelatedtotheFourierintegralofslowlygrowingfunctions.ThepartoftheFourierex-ponentialsherewithplaythesocalledexponentialsplinesbySchoenberg.Theintegralrepresentationprovidesaflexibletoolfordealingwiththegrowingequidistantsplines.First.itallowsustocon-structarichlibraryofsplinespossessingthepropertythattranslationsofanysuchsplineformaba-sisofcorrespondingsplinespace.Itisshownthatanysuchsplineisassociatedwithadualsplinewhosetranslationsformabiorthogonalbasis.Asexampleswepresentsolutionsoftheproblemsofprojectionofagrowingfunctionontosplinespacesandofsplineinterpolationofgrowingfunc-tion.Wederiveformulasforapproximateevaluationofsplinesprojectingafunctionontothesplinespaceandestablishtherewithexactestimationsoftheapproximationerrors.
