Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators  

作　　者：Hong-Lin Liao Tao Tang Tao Zhou 

机构地区：[1]School of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China [2]Division of Science and Technology,BNU-HKBU United International College,Zhuhai 519087,China [3]SUSTech International Center for Mathematics,Shenzhen 518055,China [4]NCMIS&LSEC,Institute of Computational Mathematics and Scientific/Engineering Computing,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China

出　　处：《Science China Mathematics》2024年第2期237-252,共16页中国科学（数学）（英文版）

基　　金：Hong-Lin Liao was supported by National Natural Science Foundation of China(Grant No.12071216);Tao Tang was supported by Science Challenge Project(Grant No.TZ2018001);National Natural Science Foundation of China(Grants Nos.11731006 and K20911001);Tao Zhou was supported by National Natural Science Foundation of China(Grant No.12288201);Youth Innovation Promotion Association(CAS);Henan Academy of Sciences.

摘　　要：The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

关 键 词：discrete convolution kernels positive definiteness variable time-stepping orthogonal convolution kernels complementary convolution kernels 

分 类 号：O177[理学—数学]