id	sid	eid	entity	type
ma-382	1	1	2025	CARDINAL
ma-382	2	1	j. math	PERSON
ma-382	4	1	5 (	CARDINAL
ma-382	4	2	382doi	CARDINAL
ma-382	4	3	10.28924	CARDINAL
ma-382	4	4	robert obogi2,∗ 1department	PERSON
ma-382	4	5	jaramogi	ORG
ma-382	4	6	odinga university of science and technology	ORG
ma-382	4	7	kenya	GPE
ma-382	5	1	2department	TIME
ma-382	5	2	kisii university	ORG
ma-382	5	3	kenya	GPE
ma-382	5	4	robogi@kisiiuniversity.ac.ke	ORG
ma-382	8	1	1	CARDINAL
ma-382	8	2	bounds;(2	PERSON
ma-382	8	3	3	CARDINAL
ma-382	8	4	4	CARDINAL
ma-382	9	1	computable versionsand	ORG
ma-382	10	1	1	CARDINAL
ma-382	11	1	2	CARDINAL
ma-382	12	1	lin-denstrauss	PERSON
ma-382	12	2	12	CARDINAL
ma-382	12	3	3	CARDINAL
ma-382	12	4	4	CARDINAL
ma-382	12	5	13	CARDINAL
ma-382	13	1	three	CARDINAL
ma-382	13	2	5	CARDINAL
ma-382	13	3	1	CARDINAL
ma-382	13	4	14 may 2025	DATE
ma-382	15	1	pde optimization	ORG
ma-382	16	1	j. math	PERSON
ma-382	18	1	10.28924	CARDINAL
ma-382	18	2	10	CARDINAL
ma-382	19	1	6	CARDINAL
ma-382	19	2	8	DATE
ma-382	19	3	1-2	CARDINAL
ma-382	20	1	11	CARDINAL
ma-382	20	2	2	CARDINAL
ma-382	21	1	4	CARDINAL
ma-382	21	2	geo-metric	PERSON
ma-382	21	3	3	CARDINAL
ma-382	21	4	6	CARDINAL
ma-382	21	5	12	CARDINAL
ma-382	23	1	2	CARDINAL
ma-382	26	1	〉	CARDINAL
ma-382	26	2	∈	ORG
ma-382	26	3	∈	ORG
ma-382	27	1	1	CARDINAL
ma-382	28	1	0	CARDINAL
ma-382	28	2	0	CARDINAL
ma-382	31	1	2	CARDINAL
ma-382	31	2	1− δ(ε	WORK_OF_ART
ma-382	33	1	2	CARDINAL
ma-382	37	1	3	CARDINAL
ma-382	38	1	∈	ORG
ma-382	38	2	x0	ORG
ma-382	39	1	1	CARDINAL
ma-382	41	1	j. math	PERSON
ma-382	43	1	10.28924	CARDINAL
ma-382	43	2	3	CARDINAL
ma-382	44	1	4	CARDINAL
ma-382	45	1	∑n k=1 akek‖	PERSON
ma-382	46	1	5	CARDINAL
ma-382	47	1	∈	ORG
ma-382	47	2	n ∈ n	ORG
ma-382	47	3	2−n	CARDINAL
ma-382	48	1	6	CARDINAL
ma-382	49	1	•	CARDINAL
ma-382	49	2	•	CARDINAL
ma-382	49	3	7	CARDINAL
ma-382	50	1	∈	ORG
ma-382	50	2	⊂	GPE
ma-382	54	1	3	CARDINAL
ma-382	54	2	2	CARDINAL
ma-382	56	1	1	CARDINAL
ma-382	56	2	2−n	CARDINAL
ma-382	56	3	2	CARDINAL
ma-382	57	1	− fn‖x∗	ORG
ma-382	57	2	3	CARDINAL
ma-382	57	3	hilbert spaces	ORG
ma-382	62	1	∑ α	PERSON
ma-382	63	1	1	CARDINAL
ma-382	63	2	2−n	CARDINAL
ma-382	66	1	x∗.	PERSON
ma-382	68	1	j. math	PERSON
ma-382	70	1	10.28924	CARDINAL
ma-382	70	2	4search	CARDINAL
ma-382	71	1	0	CARDINAL
ma-382	72	1	2	CARDINAL
ma-382	78	1	8	CARDINAL
ma-382	80	1	− fn‖x∗	ORG
ma-382	81	1	3	CARDINAL
ma-382	82	1	∈	ORG
ma-382	82	2	dimxn = n	PERSON
ma-382	83	1	1	CARDINAL
ma-382	83	2	1	CARDINAL
ma-382	83	3	2	CARDINAL
ma-382	83	4	sup‖x‖=1	PERSON
ma-382	85	1	3	CARDINAL
ma-382	85	2	α(p	PERSON
ma-382	89	1	∈	ORG
ma-382	89	2	bea	ORG
ma-382	91	1	1	CARDINAL
ma-382	92	1	zero	CARDINAL
ma-382	93	1	2	CARDINAL
ma-382	94	1	‖x‖ = 1	PRODUCT
ma-382	96	1	ω(f	PERSON
ma-382	97	1	3	CARDINAL
ma-382	97	2	1	CARDINAL
ma-382	97	3	α(p	PERSON
ma-382	97	4	α(p	PERSON
ma-382	101	1	j. math	PERSON
ma-382	103	1	10.28924	CARDINAL
ma-382	103	2	5	CARDINAL
ma-382	103	3	4	CARDINAL
ma-382	104	1	∈	ORG
ma-382	104	2	x0	ORG
ma-382	104	3	x0	ORG
ma-382	105	1	∈	ORG
ma-382	105	2	1	CARDINAL
ma-382	106	1	2	CARDINAL
ma-382	106	2	1− ε ‖f‖ ≤ ‖g‖	PRODUCT
ma-382	106	3	3	CARDINAL
ma-382	109	1	∈	ORG
ma-382	109	2	x0	ORG
ma-382	109	3	x0	ORG
ma-382	109	4	∈	ORG
ma-382	110	1	1	CARDINAL
ma-382	111	1	2x ∗	QUANTITY
ma-382	111	2	j(x	ORG
ma-382	112	1	∈	ORG
ma-382	113	1	∈	ORG
ma-382	113	2	x0	ORG
ma-382	115	1	2	CARDINAL
ma-382	115	2	− y‖	PERSON
ma-382	118	1	2ε	CARDINAL
ma-382	120	1	g(xε	ORG
ma-382	121	1	x0	ORG
ma-382	122	1	x0)−	NORP
ma-382	123	1	x0)−	NORP
ma-382	123	2	− xε‖.	PERSON
ma-382	124	1	2	CARDINAL
ma-382	125	1	ε⇒	NORP
ma-382	125	2	1− ε ‖f‖	PRODUCT
ma-382	125	3	1 +	CARDINAL
ma-382	126	1	3	CARDINAL
ma-382	129	1	0	DATE
ma-382	129	2	lim	PERSON
ma-382	132	1	j. math	PERSON
ma-382	134	1	10.28924	CARDINAL
ma-382	134	2	6	CARDINAL
ma-382	134	3	5	CARDINAL
ma-382	134	4	1	CARDINAL
ma-382	134	5	2	CARDINAL
ma-382	134	6	ω(n	DATE
ma-382	134	7	3	CARDINAL
ma-382	134	8	2	CARDINAL
ma-382	136	1	∈	ORG
ma-382	136	2	2−n	CARDINAL
ma-382	137	1	1	CARDINAL
ma-382	142	1	max x∈s(rn	PERSON
ma-382	142	2	2−k	CARDINAL
ma-382	142	3	s(rn	ORG
ma-382	143	1	2	CARDINAL
ma-382	143	2	ω(n	DATE
ma-382	144	1	xn	ORG
ma-382	144	2	ω(n	GPE
ma-382	145	1	3	CARDINAL
ma-382	147	1	2	CARDINAL
ma-382	148	1	2	CARDINAL
ma-382	149	1	6	CARDINAL
ma-382	151	1	∈	ORG
ma-382	151	2	1	CARDINAL
ma-382	151	3	‖fn‖	NORP
ma-382	151	4	2	CARDINAL
ma-382	151	5	3	CARDINAL
ma-382	152	1	e∗n	DATE
ma-382	154	1	n∑ k=1	PERSON
ma-382	157	1	j. math	PERSON
ma-382	159	1	10.28924	CARDINAL
ma-382	160	1	∈	ORG
ma-382	162	1	1	CARDINAL
ma-382	162	2	e∗k}nk=1	DATE
ma-382	162	3	hahn	PERSON
ma-382	163	1	‖xn‖	ORG
ma-382	163	2	1	CARDINAL
ma-382	163	3	2	CARDINAL
ma-382	164	1	n∑ k=1	PERSON
ma-382	166	1	1≤k≤n	CARDINAL
ma-382	169	1	0	CARDINAL
ma-382	170	1	ku	PERSON
ma-382	173	1	3	CARDINAL
ma-382	173	2	1	CARDINAL
ma-382	173	3	∈	ORG
ma-382	173	4	1	CARDINAL
ma-382	173	5	1	CARDINAL
ma-382	174	1	1.write	CARDINAL
ma-382	176	1	n∑ k=1	PERSON
ma-382	176	2	n∑ k=1	PERSON
ma-382	177	1	‖fn‖	LOC
ma-382	180	1	7	CARDINAL
ma-382	180	2	minu∈u j(u	PERSON
ma-382	180	3	1	CARDINAL
ma-382	181	1	j. math	PERSON
ma-382	183	1	10.28924	CARDINAL
ma-382	183	2	2	CARDINAL
ma-382	183	3	euler	PERSON
ma-382	183	4	3	CARDINAL
ma-382	187	1	1	CARDINAL
ma-382	187	2	∈	ORG
ma-382	188	1	j(u	PERSON
ma-382	189	1	coercive.then j	PERSON
ma-382	190	1	2	CARDINAL
ma-382	190	2	euler-lagrange	ORG
ma-382	190	3	jh = j|xh	PERSON
ma-382	191	1	∈	ORG
ma-382	191	2	euler-lagrange	ORG
ma-382	191	3	j ′h(uh)(v	PERSON
ma-382	191	4	xh.by cea	PERSON
ma-382	191	5	j ′′	PERSON
ma-382	191	6	0	CARDINAL
ma-382	191	7	3	CARDINAL
ma-382	196	1	second	ORDINAL
ma-382	197	1	1	CARDINAL
ma-382	197	2	2	CARDINAL
ma-382	199	1	8	CARDINAL
ma-382	200	1	∈	ORG
ma-382	200	2	1	CARDINAL
ma-382	200	3	f̂n	ORG
ma-382	200	4	2	CARDINAL
ma-382	200	5	3	CARDINAL
ma-382	202	1	∈	ORG
ma-382	203	1	zero	CARDINAL
ma-382	205	1	j. math	PERSON
ma-382	207	1	10.28924	CARDINAL
ma-382	207	2	9risk	CARDINAL
ma-382	208	1	f̂n	ORG
ma-382	208	2	xi}ni=1.in	ORDINAL
ma-382	208	3	supremum	ORG
ma-382	208	4	f̂n	ORG
ma-382	209	1	f̂n	ORG
ma-382	210	1	− sup	PERSON
ma-382	211	1	f̂n	ORG
ma-382	212	1	7→	CARDINAL
ma-382	213	1	∈	ORG
ma-382	215	1	f̂n	ORG
ma-382	217	1	9	CARDINAL
ma-382	218	1	∈	ORG
ma-382	218	2	1	CARDINAL
ma-382	218	3	2	CARDINAL
ma-382	218	4	3	CARDINAL
ma-382	218	5	4	CARDINAL
ma-382	218	6	x∗u	GPE
ma-382	219	1	1	CARDINAL
ma-382	219	2	2	CARDINAL
ma-382	219	3	1	CARDINAL
ma-382	219	4	x0	ORG
ma-382	220	1	x0	ORG
ma-382	221	1	x0	PERSON
ma-382	222	1	weak∗exposed	NORP
ma-382	223	1	2)⇒	CARDINAL
ma-382	223	2	3	CARDINAL
ma-382	223	3	x0	PERSON
ma-382	224	1	∈	ORG
ma-382	225	1	j. math	PERSON
ma-382	227	1	10.28924	CARDINAL
ma-382	227	2	10 x0	QUANTITY
ma-382	227	3	x0	ORG
ma-382	228	1	x0	ORG
ma-382	230	1	3	CARDINAL
ma-382	230	2	4	CARDINAL
ma-382	230	3	xu	PERSON
ma-382	230	4	1	CARDINAL
ma-382	232	1	4	CARDINAL
ma-382	232	2	1	CARDINAL
ma-382	233	1	xu	PERSON
ma-382	233	2	xu	PERSON
ma-382	233	3	1	CARDINAL
ma-382	237	1	4	CARDINAL
ma-382	238	1	1)constructive	CARDINAL
ma-382	238	2	1-2	CARDINAL
ma-382	238	3	2	CARDINAL
ma-382	238	4	3-4),and	CARDINAL
ma-382	238	5	3	CARDINAL
ma-382	238	6	6	CARDINAL
ma-382	239	1	5	CARDINAL
ma-382	240	1	2	CARDINAL
ma-382	240	2	12	CARDINAL
ma-382	240	3	•	CARDINAL
ma-382	240	4	•	CARDINAL
ma-382	240	5	3	CARDINAL
ma-382	240	6	•	CARDINAL
ma-382	240	7	•	CARDINAL
ma-382	240	8	•	CARDINAL
ma-382	240	9	•	CARDINAL
ma-382	240	10	james theorem	PERSON
ma-382	240	11	settingsthese	NORP
ma-382	242	1	1	CARDINAL
ma-382	242	2	acosta	PERSON
ma-382	242	3	f.j. aguirre	PERSON
ma-382	242	4	r. paya	PERSON
ma-382	242	5	israel	GPE
ma-382	243	1	154	CARDINAL
ma-382	243	2	2006	DATE
ma-382	243	3	115–131.[2	CARDINAL
ma-382	243	4	e. bishop	PERSON
ma-382	243	5	r.r. phelps	PERSON
ma-382	244	1	amer	PERSON
ma-382	246	1	67	CARDINAL
ma-382	246	2	97–98.[3	CARDINAL
ma-382	246	3	j. bourgain	PERSON
ma-382	246	4	israel	GPE
ma-382	247	1	28	CARDINAL
ma-382	247	2	1977	DATE
ma-382	247	3	265–271	DATE
ma-382	250	1	j. math	PERSON
ma-382	252	1	10.28924	CARDINAL
ma-382	252	2	11	CARDINAL
ma-382	253	1	4	CARDINAL
ma-382	254	1	s.b. cooper	PERSON
ma-382	254	2	a. sorbi	PERSON
ma-382	254	3	carothers	ORG
ma-382	254	4	cambridge university press	ORG
ma-382	254	5	2000).[6	CARDINAL
ma-382	254	6	m.n. evans	ORG
ma-382	254	7	a. samwel o.	PERSON
ma-382	255	1	curr	PERSON
ma-382	258	1	25	CARDINAL
ma-382	258	2	2025	DATE
ma-382	258	3	287–294.[7	CARDINAL
ma-382	258	4	m.n. evans	ORG
ma-382	258	5	moraa	ORG
ma-382	258	6	asian	NORP
ma-382	258	7	j. adv	PERSON
ma-382	260	1	19	CARDINAL
ma-382	260	2	r. obogi	PERSON
ma-382	262	1	8 (	CARDINAL
ma-382	262	2	2024	CARDINAL
ma-382	263	1	79–88.[9	CARDINAL
ma-382	263	2	m.n. evans	ORG
ma-382	263	3	r. obogi	PERSON
ma-382	263	4	hilbert-schmidt	ORG
ma-382	263	5	ann	PERSON
ma-382	265	1	31 (	CARDINAL
ma-382	265	2	2025	DATE
ma-382	268	1	31	CARDINAL
ma-382	268	2	1987	DATE
ma-382	269	1	m. hinze	PERSON
ma-382	269	2	r. pinnau	PERSON
ma-382	269	3	m. ulbrich	PERSON
ma-382	269	4	s. ulbrich	PERSON
ma-382	269	5	pde constraints	ORG
ma-382	269	6	2009).[12	CARDINAL
ma-382	269	7	israel	GPE
ma-382	270	1	1	CARDINAL
ma-382	270	2	1963	DATE
ma-382	270	3	139–148.[13	CARDINAL
ma-382	271	1	2000	DATE
ma-382	273	1	1	CARDINAL
ma-382	273	2	2	CARDINAL
ma-382	273	3	3	CARDINAL
ma-382	273	4	4	CARDINAL