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Estimations in \(L_q\)-norm of the difference between spectral functions of the Gegenbauer-type operators. (English. Russian original) Zbl 1011.47031
Russ. Math. 43, No. 8, 17-22 (1999); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1999, No. 8, 20-25 (1999).
Let \(L_q\) be the space of complex-valued functions on \([-1,1]\) with the norm \[ \|f\|_{L_q}= \left(\int^1_{-1} \bigl|f(x)\bigr |^q (1-x^2)^{\gamma- 1/2}ds \right)^{1/p}, \quad 1\leq q<\infty,\;0<\gamma <1. \] Let \(T\) be the operator defined on \(L_q\) by \(Ty=-(1-x^2)y''+ (2\gamma+1) xy'\) and \(v_n(x)=c_n C^\gamma_n(x)\) be its eigenfunctions orthonormed in \(L_2\) corresponding to the eigenvalues \(\lambda_n=n (n+2\gamma)\), where \(c_n\) are constants and \(C_n^\gamma (x)\) are the Gegenbauer polynomials. Let \(P: L_2 \to L_2\) be the operator of the multiplication by a function \(p\in L_\infty\) (real-valued measurable and essentially bounded functions on \([-1,1])\). Denote by \(\mu_n\) the eigenvalues of \(T+P\) which are enumerated in order of increasing real parts and \(u_n\) means the system of corresponding orthonormed eigenfunctions. In the paper the author considers properties of the functions \(\sum^n_{j=1}u_j(x) \overline {u_j(y)}-\sum^n_{j=1} v_j(x)\overline {v_j(y)}\) and gives estimations of their \(L_q\)-norm \((1\leq q\leq 2)\).
MSC:
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34L05 General spectral theory of ordinary differential operators
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)