Fractional Jacobi Renormalization Operators for Fixed Classical Orthogonal Polynomial Systems

For a fixed orthogonal polynomial sequence, classical orthonormalization changes the polynomials by degree-dependent scalar factors. If the polynomials themselves are kept fixed, scalar-measure normalization is usually impossible; on a compact interval this follows from the standard uniqueness of the orthogonality measure. This paper studies the inverse Hilbert-space problem: given a prescribed classical polynomial sequence, which operator-induced geometry makes it orthonormal? For Jacobi polynomials, this geometry is described, in a precise quadratic-form sense, by a Jacobi Hilbert scale. More precisely, when the prescribed sequence is obtained from Jacobi polynomials by scalar factors whose size grows like a fixed power of the degree, the unique diagonal renormalizing operator is a Borel spectral multiplier of the Jacobi differential operator, and its eigenvalue order determines the Hilbert-scale completion of the polynomial space. The resulting normalization index records the Hilbert-scale order forced by the fixed normalization. Once this index is known, boundedness, compactness, and Schatten-class criteria follow from standard diagonal operator theory. The same index also yields a spectral-truncation estimate in the induced norm and identifies the modal filter arising from Tikhonov regularization with a Euclidean coefficient penalty. Applications to standard Jacobi, Legendre, Gegenbauer, and Chebyshev systems are obtained, including the bounded edge cases represented by the Chebyshev polynomials of the first and second kinds. Numerical tables illustrate the predicted spectral slopes, the exact Legendre identity, and the Gegenbauer Schatten threshold.