TY - JOUR
T1 - On the multiplicities of the zeros of laguerre-pólya functions
AU - Kamimoto, Joe
AU - Ki, Haseo
AU - Kim, Young One
PY - 2000/12/1
Y1 - 2000/12/1
N2 - We show that all the zeros of the Fourier transforms of the functions exp(-x2m), m = 1,2,⋯, are real and simple. Then, using this result, we show that there are infinitely many polynomials p(x1,⋯, xn) such that for each (m1,⋯, mn) ∈ (ℕ \ {0})n the translates of the function p(x1,⋯, xn)exp (-∑j=1nxj2mj) generate L1(ℝn). Finally, we discuss the problem of finding the minimum number of monomials pα(x1,⋯, xn), α ∈ A, which have the property that the translates of the functions pα(x1,⋯, xn)exp(-∑j=1nxj2mj), α ∈ A, generate L1ℝn), for a given (m1,⋯,mn) ∈ (ℕ\{0})n.
AB - We show that all the zeros of the Fourier transforms of the functions exp(-x2m), m = 1,2,⋯, are real and simple. Then, using this result, we show that there are infinitely many polynomials p(x1,⋯, xn) such that for each (m1,⋯, mn) ∈ (ℕ \ {0})n the translates of the function p(x1,⋯, xn)exp (-∑j=1nxj2mj) generate L1(ℝn). Finally, we discuss the problem of finding the minimum number of monomials pα(x1,⋯, xn), α ∈ A, which have the property that the translates of the functions pα(x1,⋯, xn)exp(-∑j=1nxj2mj), α ∈ A, generate L1ℝn), for a given (m1,⋯,mn) ∈ (ℕ\{0})n.
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M3 - Article
AN - SCOPUS:22844456227
SN - 0002-9939
VL - 128
SP - 189
EP - 194
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 1
ER -