Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let B"n:={b"0,b"1,...,b"n}(n=0,1,...,N;N@?N) be the sets of linearly independent functions.  give a simple method of construction, the dual functions D"n:={d"0^(^n^),d"1^(^n^),...,d"n^(^n^)}(0@?n@?N) satisfying the following conditions: spanD"n=spanB"n and =@d"i"j(0@?i,j@?n@?N), where @d"i"i=1, @d"i"j=0 for i j, and   is a given inner product. The proposed algorithm allows us to construct all the sets of the dual functions D"0,D"1,...,D"N in the time O(N^3), where N is a natural number. Four illustrative examples presenting the possible applications of obtained results are given.
['Paweł Wony']
I am writing a research paper on a idea Construction of dual bases
[{"from": "human", "value": "I am writing a research paper on a idea Construction of dual bases"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let B\"n:={b\"0,b\"1,...,b\"n}(n=0,1,...,N;N@?N) be the sets of linearly independent functions.  give a simple method of construction, the dual functions D\"n:={d\"0^(^n^),d\"1^(^n^),...,d\"n^(^n^)}(0@?n@?N) satisfying the following conditions: spanD\"n=spanB\"n and =@d\"i\"j(0@?i,j@?n@?N), where @d\"i\"i=1, @d\"i\"j=0 for i j, and   is a given inner product. The proposed algorithm allows us to construct all the sets of the dual functions D\"0,D\"1,...,D\"N in the time O(N^3), where N is a natural number. Four illustrative examples presenting the possible applications of obtained results are given."}]