A note on differential operators of infinite order

Cha, Young-Joon; Ki, Haseo; Kim, Young-One

doi:10.1016/j.jmaa.2003.10.033

Cited by 8 publications

(6 citation statements)

References 3 publications

(3 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, φ(D) m−1 f ∈ dom φ(D). It is obvious that if f is a polynomial, then f ∈ dom φ(D) m for all m. For more general restrictions on the growth of φ and f under which f ∈ dom φ(D) m for all m, see [4,6].…”

Section: Introductionmentioning

confidence: 99%

“…(2) From [6, Lemma 3.2], [11,Theorem 2.3] and the arguments given in [4], it follows that the restriction "f is of order less than 2" can be weakened to "φ or f is of genus at most 1". See also [6,Theorem 3.3].…”

Section: Introductionmentioning

confidence: 99%

“…(3) In the case where φ is of genus 2, that is, φ is of the form φ(x) = e −γx 2 ψ(x), where γ > 0 and ψ ∈ LP is of genus at most 1, we have the following stronger result: If f is a real entire function of genus at most 1, and if the imaginary parts of the zeros of f are uniformly bounded, then f ∈ dom φ(D) m and Z C (φ(D) m f ) ≥ Z C (φ(D) m+1 f ) for all m, and Z C (φ(D) m f ) → 0 as m → ∞, even when f has infinitely many nonreal zeros. See [3, Theorems 9a, 13 and 14], [4], [6,Lemma 3.2] and [11,Theorem 2.3].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Department of Chemistry, Seoul National University

Kim

2008

Mol. Sci.

View full text Add to dashboard Cite

I . I n t r o d u c t i o nT h i s s h o r t a r t i c l e i s a c o n t r i b u t i o n t o t h e we b j o u r n a l " Mo l e c u l a r S c i e n c e " i n a n e f f o r t t o i n v i g o r a t e a n d d e e p e n mu t u a l u n d e rs t a n d i n g a mo n g v a r i o u s i n s t i t u t i o n s i n t h e a r e a o f mo l e c u l a r s c i e n c e i n d i f f e r e n t c o u n t r i e s , e s p e c i a l l y i n t h e E a s t As i a n r e g i o n o f t h e wo r l d . I n t h e p r e s e n t i s s u e o f t h e j o u r n a l , t h e De p a r t me n t o f C h e mi s t r y o f S e o u l Na t i o n a l Un i v e r s i t y i s b e i n g p r e s e n t e d i n a s e t o f v i g n e t t e s t h a t s u mma r i z e i t s h i s t o r y , o r g a ni z a t i o n , p e o p l e , a c t i v i t i e s , a n d r e s o u r c e s . T h e r e a d e r s a r e a s k e d t o b e a r wi t h t h e c o l l o q u i a l n a t u r e o f t h i s a r t i c l e e mi n e n t l y d i s p l a y e d i n s o me p l a c e s s i n c e i t i s s o d e s i g n e d t o c o n v e y a mo r e d i r e c t a n d p e r s o n a l p o i n t o f v i e w . E s t a b l i s h e d i n 1 9 4 6 , t h e De p a r t me n t o f C h e mi s t r y h a s a l o n g a n d d i s t i n g u i s h e d t r a d i t i o n i n c h e mi c a l e d u c a t i o n a n d r e s e a r c h i n Ko r e a . T h e a l u mn i a r e p l a y i n g l e a d i n g r o l e s i n a c a d e mi a , g o v e r n me n t a n d c o r p o r a t e r e s e a r c h i n s t i t u t e s , a n d ma n y a r e a s o f c h e mi c a l i n d u s t r y .At p r e s e n t 3 5 f a c u l t y me mb e r s a r e a c t i v e l y e n g a g e d i n e d u c at i o n a n d r e s e a r c h i n a l l a r e a s o f b a s i c a n d a p p l i e d c h e mi s t r y , wi t h 1 9 1 u n d e r g r a d u a t e s t u d e n t s wi t h ma j o r s i n c h e mi s t r y , 1 2 8 Ma s t e r ' s d e g r e e s t u d e n t s , 1 4 0 d o c t o r a l s t u d e n t s , 1 2 p o s t d o c t o r a l r e s e a r c h e r s , a n d s e v e r a l me mb e r s o f t h e t e a c h i n g a n d r e s e a r c h s t a f f . C o l l a b o r a t i o n wi t h f a c u l t y i n p h y s i c s , l i f e s c i e n c e s , me d i c in e , a n d ma t e r i a l s c i e n c e wi t h i n a n d o u t s i d e t h e u n i v e r s i t y i s a c t i v e l y p u r s u e d .T h e De p a r t me n t h a s b e e n s u p p o r t e d b y t h e B K2 1 P r o g r a m, a mu l t i -p h a s e , 7 -y e a r g o v e r n me n t p r o j e c t t h a t p r o v i d e s f i n a n c i a l s u p p o r t t o g r a d u a t e s t u d e n t s a n d p o s t d o c t o r a l r e s e a r c h e r s . E x t e rn a l l y f u n d e d r e s e a r c h c e n t e r s a n d i n s t i t u t e s a s we l l a s v a r i o u s c e n t e r s o f e x c e l l e n c e a r e a l s o h o s t e d b y t h e De p a r t me n t .T h e a c a d e mi c y e a r i n K o r e a i s f r o m Ma r c h 1 s t t o t h e e n d o f F e b r u a r y o f t h e f o l l o wi n g y e a r . E a c h a c a d e mi c y e a r c o n s i s t s o f t h e s p r i n g a n d f a l ...

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Department of Chemistry, Seoul National University

Kim

2008

Mol. Sci.

View full text Add to dashboard Cite

show abstract

“…namely the order and the type of F (t, •), which for convenience we call the zorder and z-type of F (t, z). One might think that these quantities depend on t. However, by applying Theorem 1.1 of [2] for the choice f (D) = e tD 2 (in the notation of Theorem 1.1 of [2]) and noticing that F (t, z) = e tD 2 F (0, z) with D = ∂ z , we get that ρ z and τ z are, actually, independent of t (actually, this also follows from (118)). We can, therefore, choose t = 0 and conclude that…”

Section: 3mentioning

confidence: 99%

Analytic Solutions of the Heat Equation

Papanicolaou¹,

Kallitsi²,

Smyrlis³

2019

Preprint

View full text Add to dashboard Cite

Motivated by the recent proof of Newman's conjecture [12] we study certain properties of entire caloric functions, namely solutions of the heat equation ∂tF = ∂ 2 z F which are entire in z and t. As a prerequisite, we establish some general properties of the order and type of an entire function. Then, we start our inquiry on entire caloric functions by determining the necessary and sufficient condition for a function f (z) to be the initial condition of an entire solutions of the heat equation and, subsequently, we examine the relation of the z-order and z-type of an entire caloric function F (t, z), viewed as function of z, to its t-order and t-type respectively, if it is viewed as function of t. After that, we shift our attention to the zeros z k (t) of an entire caloric function F (t, z), viewed as function of z. We show that the points (t, z) at which F (t, z) = ∂zF (t, z) = 0 form a discrete set in C 2 and we derive the t-evolution equations of the zeros of F (t, z). These are differential equations which hold for all but countably many t ∈ C.

show abstract

“…In particular, according to these papers, in this case it seems to be easier to prove the applicability of such operators to certain subsets of the space of entire functions. For example, in [2] Cha et al have been able to provide a sufficient condition for two entire functions, f pzq and gpzq, to be such that the combination ř n f pnq p0q D n gpzq{n! represents an entire function.…”

Section: 1mentioning

confidence: 99%