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Fixing a basepoint $o \\in X$, for $\\xi \\in \\partial X$, let $B_{\\xi}$ denote the Busemann function at $\\xi$ such that $B_{\\xi}(o) = 0$, then for $\\lambda \\in \\mathbb{C}$ the function $e^{(i\\lambda - \\rho)B_{\\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\\lambda^2 + \\rho^2)$.\n  For a function $f$ on $X$, we defi"},"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":"1802.07236","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-02-20T18:30:25Z","cross_cats_sorted":[],"title_canon_sha256":"0a17d790c0f40756df67c432a325c680c362e1747b60b29a3633b6a3165c8021","abstract_canon_sha256":"5e71c94d57778f4a13b1252a9254d452a2c167da22a7201988a46dce65c2d6dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:50.559847Z","signature_b64":"wft2KVBC+n6nwVBWqRZ/LUX6/5IIjYrQkQySEOZEZpN4H7iuEwNX7S+NEglzqZHlzO1oNFq6g9UnHzvwqZZzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62a6faf5fa93de0fff6313530992b143a38b9e6cb9fe33fabf1e81ea0829f047","last_reissued_at":"2026-05-18T00:22:50.559462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:50.559462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Fourier transform on negatively curved harmonic manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kingshook Biswas","submitted_at":"2018-02-20T18:30:25Z","abstract_excerpt":"Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \\leq -1$, and let $\\partial X$ denote the boundary at infinity of $X$. 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