On Summability of Spectral Expansions Corresponding to the Sturm-Liouville Operator

Table 1

Case	Class	Conditions on the 𝐴𝑖𝑗	Completeness	Basis property
(1)	SR	𝐴12≠0; 𝐴12=0, 𝐴14+𝐴23≠0, 𝐴14+𝐴23≠∓(𝐴13+𝐴24); 𝐴12=0, 𝐴14+𝐴23=0, 𝐴13+𝐴24=0, 𝐴13=𝐴24, 𝐴34≠0	Yes	Yes
(2a)	WR	𝐴12=0, 𝐴14+𝐴23≠0, 𝐴14+𝐴23=∓(𝐴13+𝐴24), 𝐴14=𝐴23, 𝐴34≠0	Yes	Yes
(2b)	WR	𝐴12=0, 𝐴14+𝐴23≠0, 𝐴14+𝐴23=∓(𝐴13+𝐴24), 𝐴14=𝐴23, 𝐴34=0; 𝐴12=0, 𝐴14+𝐴23≠0, 𝐴14+𝐴23=∓(𝐴13+𝐴24), 𝐴14≠𝐴23	Yes	Yes/No
(3)	IR	𝐴12=0, 𝐴14+𝐴23=0, 𝐴13+𝐴24=0, 𝐴13≠𝐴24, 𝐴34≠0; 𝐴12=0, 𝐴14+𝐴23=0, 𝐴13+𝐴24≠0,𝐴34≠0	Yes	No
(4)	DEG	𝐴12=0, 𝐴14+𝐴23=0, 𝐴34=0	Yes/No	?/No