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We show that the FL represents the \"{\\it fractional continuum limit}\" of a discrete \"self-similar Laplacian\" which is obtained by Hamilton's variational principle from a discrete spring model.\nWe deduce from generalized self-similar elastic potentials regular representations for the FL which involve convolutions of symmetric finite difference operators of even orders extending the standard representation of the FL.\nFurther we deduce "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.01942","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-01-08T20:07:25Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP"],"title_canon_sha256":"c69dd768e8e5cd4578b69cc6f5fbdcaa47b5a2e6866105920cd861ddf3b0744a","abstract_canon_sha256":"dd47fe3b6de838fb7011723041fa7d56ff85fb4700ef37170fe30d84123772b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:53.683908Z","signature_b64":"WWykqSOqQfAjqBvBgt6LS3I6ck6M2JTSp4/6FNxfxtPVSJ2RoWZTKcGf42krPfE/in4r1a/qX4dMq/X0D7DlBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"408f4a77d55a33488c02aadc927c3b818c559cd1539cbc9e4cddfcb1fc8b5a71","last_reissued_at":"2026-05-18T02:28:53.683506Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:53.683506Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot type fractal functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP"],"primary_cat":"math-ph","authors_text":"G\\'erard Maugin (IJLRA), Rahman Mujibur, Shahram Derogar (MACE), Thomas Michelitsch (IJLRA)","submitted_at":"2015-01-08T20:07:25Z","abstract_excerpt":"We demonstrate that the fractional Laplacian (FL) is the principal characteristic operator of harmonic systems with {\\it self-similar}\ninterparticle interactions. 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