# zbMATH — the first resource for mathematics

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.)
##### Keywords:
spectral function; eigenfunctions; Gegenbauer polynomials