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These functions are eigenfunctions of the $p$-adic pseudo-differential Vladimirov operator, which is defined on a compact set $B_{r}\\subset\\mathbb{Q}_{p}$ of the field of $p$-adic numbers $\\mathbb{Q}_{p}$ or, respectively, on the entire field $\\mathbb{Q}_{p}$. A relation between the basis of functions from $L^{2}\\left(\\mathbb{Q}_{p}\\right)$ and the basis of $p$-adic wavelets from $L^{2}\\left(\\mathbb{Q}_{p}\\right)$ is found. As an application, we consider the solution of the Cauc"},"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":"1504.03624","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-04-14T16:54:59Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"8ca434db9474acff0219c26cacd844c946f9bebfdf793ab59967ddd253bddbe4","abstract_canon_sha256":"00441391df7157014e6b8528a399b6a3ef0ff575c75f27c0e3007d710a91145e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:50.360770Z","signature_b64":"nPBpaXn+Fs9/gW5a7vFwAlyB6dLKHWbOiLRHX7AvE0EI+yviL/6NwGAcw3YSgbsI6aLCmFRyp0GXEbUtQ9kvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"641e33d995fd971392d90ecbc2f3ce39eba250f3daaf7abd7851212e7721650b","last_reissued_at":"2026-05-18T02:18:50.360192Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:50.360192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On one real basis for $L^2(Q_p)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"A.Kh. Bikulov, A.P. Zubarev","submitted_at":"2015-04-14T16:54:59Z","abstract_excerpt":"We construct new bases of real functions from $L^{2}\\left(B_{r}\\right)$ and from $L^{2}\\left(\\mathbb{Q}_{p}\\right)$. These functions are eigenfunctions of the $p$-adic pseudo-differential Vladimirov operator, which is defined on a compact set $B_{r}\\subset\\mathbb{Q}_{p}$ of the field of $p$-adic numbers $\\mathbb{Q}_{p}$ or, respectively, on the entire field $\\mathbb{Q}_{p}$. A relation between the basis of functions from $L^{2}\\left(\\mathbb{Q}_{p}\\right)$ and the basis of $p$-adic wavelets from $L^{2}\\left(\\mathbb{Q}_{p}\\right)$ is found. 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